TH THOMIST A SPECULATIVE QUARTERLY REVIEW OF THEOLOGY AND PHILOSOPHY EDITORS: THE DoMINICAN FATHERS OF THE PRoVINCE OF ST. JosEPH Publishers: The Thomist Press, Washington 17, D. C. VoL. XV JULY, 195£ No.3 THE MORALITY OF BASING-POINT PRICING N ORDINARY consumer lives, buys and dies without ever understanding the intricacies of price determination. Supply and demand seem to be the main causes. But sometimes neither supply nor demand seem important as prices get " stuck " at a certain level, whether buyers take little or much. Sometimes the suspicion arises that sellers are following the policy of charging all that the market can bear. Politicians and employers frequently blame labor unions for rising costs and prices. Then, to complicate matters, inflation and defiation make money cheaper or dearer, which is the same as raising o:r lowering all the prices at once. Actually, pricing policies are even more complicated than most people suspect. One does pay more when buying on credit; is the higher price due to the financial accomodation, or are there " penalties " included for not buying cash? What about zoning systems? A roll of newsprint sells for one price 349 350 R..4.YMOND C. JANCAUSKAS throughout Canada, no matter where it comes from and where it arrives, while the prices in the United States change according to zones that radiate from Canada. Why must such a price system be used? This article deals with basing-point pricing, one of the more complicated systems. In its most simple form, a single-basing-point system is a practice wherein all products of a given nature are priced to all buyers in all markets as though originating at a single shipping point. Actually, however, shipments may be made from that point or from any of a multiple number of producing points. The most widely known of these systems was the practice employed in the steel industry for about 50 years prior to known as Pittsburgh Plus. Under that practice, steel was sold to buyers in all markets at f. o. b. mill prices based upon the fiction that the point of origin for all shipments was Pittsburgh .... The buyer paid, irrespective of the location of his seller's mill, the f. o. b. mill price plus the rail freight cost from Pittsburgh to the buyer's destination. Thus an Iowa buyer of steel from a Gary, Ind., mill paid the f. o. b. Pittsburgh price plus the cost of rail transportation from Pittsburgh to Iowa. This practice has long since been held illegal by the courts .... 1 By making the system a little more complicated, the steel and cement companies thought they would avoid trouble. The change was very simple: The multiple-basing-point system is distinguished from the singlebasing-point system in that it includes two or more points of origin from which transportation costs are computed, which may or may not be the actual points of shipment. 2 Since the system operates on exactly the same principles, but over many small areas instead of one big area, the Federal Trade Commission has attacked it and sought its discontinuance. One way of analyzi.ng the implications of this pricing practice would be to concentrate on its immediate objective: achieving 1 U. S. Senate, Interim Report on Study of Federal Trade Commission Pricing (Washington, 1949), pp. £-8. Policies, 8lst Congress, 1st Session, Document No. This source will be referred to as the Interim Report. "Ibid .. p. 8. THE MORALITY OF BASING-POINT PRICING 351 a single price to every buyer no matter where, or from whom, he buys. According to Prof. Machlup, The quoting of identical delivered prices is the immediate objective; that it involves price discrimination on th part of individual producers located in different regions is in a sense merely incidental to the attainment of the major objective. 3 This was also the conclusion of the Temporary National Economic Committee: Extensive hearings on basing-point systems showed that they are used in many industries as an effective device for eliminating price competition .... The elimination of such systems under existing law would involve a costly process of prosecuting separately and individually many industries, and place a heavy burden upon antitrust enforcement appropriations: We therefore recommend that the Congress enact legislation declaring such pricing systems to be illegal.4 However, this approach would not lead to conclusive results in a moral evaluation of the system because it can happen (as it did in the case of steel in 1948) that, for political reasons, producers may keep their product prices at artificially low levels. 5 When monopolistic injustice is blunted in this way, the argument against such pricing is also blunted. 6 So we must take the position that the incidental but necessary discrimination that this pricing system involves is sufficient to condemn it morally. • Fritz Machlup, The Basing-Point System (Philadelphia: The Blakiston Company, 1949), p. 170. In one of his examples, 31 sealed bids were received by the United States Navy for cold rolled or cold drawn steel on May 26, 1936, all offering $20,727.26. In another, eleven cement firms submitted sealed bids to officers of a United States Engineer Office ,on April 23, 1936; all the bids were $3.286854 per barrel. • Final RepOTt and Recommendations of the Temporary National Economic Committee (Washington, 1941), p. 33. 5 Machlup, op. cit., p. lOS, footnote. • No Catholic moralist has yet found anything wrong in the mere control of a market. Interestingly enough, some lawyers claim that, after the Aluminum (1945) and Tobacco (1946) cases, market control is illegal under Section 2 of the Sherman Act. Cf. E. W. Rostow, A National Policy for the Oil Industry (New Haven: Yale University Press, 1948), p. 129. RAYMOND C. JANCAUSKAS Furthermore, in taking this approach we must go through two lines of argument because the defenders of the system jump from one basic assumption to another in making "explanations." However, in both lines of argument, it can be proven that this practice necessarily leads to discrimination both against certain producers and certain buyers. General Moral Principles To Be Applied Our procedure, then, will be to accept one of the assumptions, to analyze the working of this pricing practice, and then to see how discrimination occurs. It goes without saying that if we find consumers getting charged for a service they are not getting or asking for, such a practice will be labeled "theft." But discrimination may not go as far as theft, and such cases will be labeled a "sin against charity" or, in plain terms, hating the neighbor· in a practical way in business. If the discrimination is severe and grave damage results to the discriminated parties, the sin against charity would be grave. For the benefit of moralists, the discussion will be continued in the accepted scholastic terminology, but there is really nothing more profound in what follows than what has just been stated on the level of principle. THE CoMPETITIVE AssuMPTION One line of argument against this pricing system is based on the assumption that the industries concerned are ordinary industries and, therefore, should charge a competitive price. Historically, the industries themselves chose this line of argument by claiming that basing-point pricing was ordinary pricing: f. o. b. plus the cost of actual freight to the buyers. Furthermore, the most elaborate defense of the practice, put out by the United States Steel Corporation, insists that " competitive forces determine the prices quoted at all destinations." 7 7 Frank Fetter, "The New Plea for Basing-Point Monoply," Quarterly JoumaJ, of Economics (October, 1937), p. 601. THE MORALITY OF BASING-POINT PRICING 353 The Basing-point System The purpose of basing-point pricing is that the producers of some product, e. g. steel and cement in the past, may know the exact price at which the product is sold to every buyer. Each producer can figure out the exact final or "delivered" price (original price plus freight and handling costs) at any destination. 8 The final price is important because, as these industries maintain, the transportation costs for these products are so high that " buyers . . . are seldom interested in its (steel's) price at any place except where they need it." 9 Competition is in terms of the final price/ 0 The procedure may be illustrated by Mr. Lumberman who knows that the United States is divided into forty-eight sales territories known as " basing-point areas." In our crude example, the sales territories coincide with the State areas; and the capital of each State is a basing point. The price of lumber at all the basing points is published in the Lumber Trade Journal, or gets around some other way. Since Mr. Lumberman does not have the rate books of all the railroads, he is generally given a compilation of "applicable" freight rates. 11 Now he is in a position to figure out the price of any amount of lumber 8 Temporary National Economic Committee Monograph No. 42, "The Basing Point Problem," 76th Congress, Srd Session (Washington: Government Printing Office, 1941), p. 81. There is a qualification involved in this laniDiage, as we shall see later. • TNEC Monograph No. 42, p. 81. 10 Of course, delivery may be by ship, barge, truck or rail, and this variation in delivery methods would make the final price somewhat vague. The usual way of getting around this difficulty is to charge for rail transportation, no matter what method of delivery is used. The cement industry forbade the sellers to permit truck deliveries (Machi up, op. cit., p. 77); the steel companies allowed certain discounts when the buyers sent their trucks to the steel producer (TNEC Monograph No. 42, p. 78). Forcing any definite method of transportation, or charging more than the transportation costs, is clearly a matter of theft. Label it "holdup." 11 These rates are generally the actual rates; but if the actual rates change, the members must still follow the most recently distributed "applicable" rates. TNEC Monograph No. 42, pp. 103-4. When actual rates are less than the applicable rates, we have a " holdup." 854 RAYMOND C. JANCAUSKAS anywhere in the United States. For example, if he wants to know the price of maple-flooring in Chicago, he takes the basing-point price at Springfield plus freight from Springfield to Chicago plus some quantity and quality " extras." 12 The price of maple-flooring in New York City is the base price at Albany plus freight from Albany to New York (even if it comes down the Hudson by barge). The price in Buffalo is the base price at Albany plus freight from Albany to Buffalo. One peculiar result of the system is that, as far as the buyer is concerned, it does not matter where the producer's mill is located. The buyer buys just as cheaply from a producer in Los Angeles as from one in Seattle, St. Paul or Nashville. This may sound absurd, hut the point was clearly illustrated during the depression of the "thirties" when some government agencies ordered from the most distant points because " they felt that this at least would contribute to the prosperity of the railroads." 18 Such " illogicality " is inherent in the system and is, ultimately, the source of its discrimination. This "illogicality " is even driving the champions of basingpoint pricing to fantastic and desperate arguments. They claim, for example, that basing-point pricing is more competitive since one has more suppliers to choose from, instead of going to the nearest one, as in f. o. b. pricing. This circumscribing of consumer choice by f. o. b. mill pricing fails to meet the first test of a competitive market. It deprives the buyer of a choice of sellers and compels him to purchase from the geographically closest supplier. Required f. o. b. mill selling would virtually necessitate each contractor acquiring his supplies from the steel mill, cement plant or producer of other commodities located closest to the site of his construction job or be unable to compete. 14 12 These "extras " would also be in the trade journal, freight rate book, weekly newsletter, or in some other handy form. There will be no further reference to them. 13 John F. Cronin, S. S., Economic Analysis and Problems (New York: American Book Company, 1945), p. 14 Interim Report, p. 10. THE MORALITY OF BASING-POINT PRICING 355 Under f. o. b. pricing A steel fabricator whose plant is not located close to a steel mill or to a large consuming area finds itself at a disadvantage both as to the cost of its raw materials and as to its competitive ability to sell in distant markets.15 One can hardly believe that fear of competition would drive our anti-monopolistic businessmen into such a mental confusion. Since useless traffic helps the railroads, they actually secured the testimony of the Deputy Chief of the Army Transportation Corps and of the chairman of the Railway Labor Executives Association to tell the Senate subcommittee how useful it would be to have a healthy railroad system in time of war. 16 However, to get on with the market terminology and the various situations that may arise, we suppose, first, that Mr. Lumberman is located at a bas·ing-point, e. g. Austin, Texas. Then the cost of maple-flooring in Houston is the price at Austin plus freight to Houston. In this case, although the basingpoint system is being followed, the procedure is fair and just if (1) the base price is fair, (2) the method of transportation is such as the buyer would choose, and (8) the freight is fair; If these conditions are met, the delivered price is fair. Everything is proper. Not all the transactions, then, that take place under this pricing system unjust. There is no discrimination and no fictitious charge in the example just given. Mr. Lumberman of Austin can face any government investigator and assert that, as far as these sales go, it is ordinary competition. We may suppose, then, secondly, that the producer is still at Austin, but finds that he needs more than Texas sales to move his stocks. He meets a friend from Tulsa and sells him a carload of lumber. The price at which he sells is the base price at Oklahoma City plus freight from Oklahoma City to Tulsa. It is a strange transaction: Austin wood is shipped from Austin to Tulsa at the price of Oklahoma City wood which might have been shipped from Oklahoma City to Tulsa. Two fictitious 16 Ibid., p. 81. 18 Ibid., p. 20. 356 RAYMOND C. JANCAUSKAS prices are operating: a fictitious base price and a fictitious freight charge. If we assume further that the base price is the same in Austin and Oklahoma City, the Austin producer clearly suffers a loss on his Oklahoma sale; for Tulsa is only one hundred miles from Oklahoma City (the freight charged to the customer) but the actual shipment is from Austin to Tulsa, about 500 miles. The Austin producer pays more in freight than he charges his customer. He "absorbs" freight; and this freight absorption causes a smaller net profit at his own mill than he gets from Texas sales. The " net profit at mill " is commonly referred to as the " mill net." And it is clear that the mill net varies inversely with the amount of freight absorption. There is a uniform mill net on all Texas sales; and a decreasing mill net the further the Austin producer reaches out of Texas for business. The fictitious base price and the fictitious freight charge are necessary, as the Austin producer sees it, because he must meet the price of Oklahoma producers. There is still nothing basically unjust in the action of the Austin producer because he can practice " freight abso:rption " even if basing-point pricing is declared illegaL If he -must enlarge his market, he may grant more favorable handling costs, share some of the freight charges, or simply lower the base price for all his customers (assuming that this is within his power). In the first two cases, we have freight absorption, or its equivalent; in the third, we have a search for more customers in the ordinary competitive way. The disadvantage in the last method lies in the " bargain " the local customers get as the producer " makes a play " for more distant customers; a greater difficulty is that this independent action of the Austin producer would upset the established market areas. 17 " Freight 17 This is the way the crude oil producers operate. The " competitive " price is considered to be at the " fringe " of the market area, and the well-head price is found by subtracting the transportation cost from the " fringe " to the well-head. An enlargement of the market is achieved by lowering the '' fringe" price and thus pushing back the oil from other competitive sources. Then, since larger transportation costs are gene.rally involved because the radius of the market area is now THE MORALITY OF BASING-POINT PRICING 357 absorption " adds distant customers without lowering prices to the local buyers and without changing the usual market areas. If one is to believe the campaign-managers for the defense of the basing-point system, they are really defending the privilege of competitors to absorb freight because freight absorption leads to varying mill nets just as basing-point pricing does when the sellers get outside of their territory. If varying mill nets are declared illegal, they claim that the prices of chewinggum, newspapers, magazine, etc. will have to be scaled according to the distance of the buyer from the producer-a most absurd consequence. The whole outcry about freight absorption is beside the point. The point at issue is basing-point pricing, not freight absorption, which is incidental to certain types of sales under a basing-point system. As Prof. Machlup suggests, basingpoint producers were anxious to " broaden the opposition " to the Federal Trade Commission in its attempt to end basingpoint pricing; so they attempted to convince "more and larger groups that they are threatened by the same policy." 18 The strategy is quite clever since few people stop to consider the economics of newspaper pricing, post-office policies, or chewinggum price practices. Fewer still know what basing-point pricing is. And even if the analogy applies (from postage stamps to steel and cement), fewer still are sharp enough to distinguish between freight absorption caused by an agreement between several producers scattered through a basing-point area and the freight absorption of a bona-fide independent single seller. Returning, then, from a consideration of this " herring," we may recall that what has been described so far as basing-point practice is morally irreprehensible. We now come to the crucial cases in which we finally suppose that Mr. Hardwood has a non-basing-point mill at Fort Worth, Texas. All his Fort longer, the well-head price is lowered even more as these costs are subtracted from the lower " fringe " price. We are not considering this pricing system in the article. 18 Machlup, op. cit., pp. 57-8. 358 RAYMOND C. JANCAUSKAS Worth customers are charged the Austin price plus freight from Austin to Fort Worth. Since the wood does not come from Austin, this is known as charging " phantom freight." Should Mr. Hardwood sell some lumber to a Dallas dealer, the difference between what he actually pays for freight (for a shipment from Fort Wmth to Dallas) and what he actually charges (for a nonexistent shipment from Austin to Dallas) is also known as "phantom freighL" Just as freight invariably leads to a higher mill net; the lower mill net is a sacrifice to meet competition, the higher mill net is a reward for not being at the basing-point. 19 We shall see later that there is some justification for this "reward." There can be a mixture of the two, as for example, in 1\fr. Hardwood's sales towards Austin, e. g. Waco; for a sale in this city, about half-way between Fort Worth and Austin will give him only one half the phantom freight he would get on sales in Fort Worth. Furthermore, since he has to pay the actual freight to Waco (from Fort Worth to Waco) this charge cancels the "phantom freight" he gains (from Austin to Waco) . Such an acceptance of lower mill nets on sales towards a basing-point mill by a non-basing-point mill is also known as « freight absorption." To the seller, the lower mill net is equivalent to a penalty for selling toward the basing-point mill; and the closer he sells to Austin, the higher his penalty. "Freight absorption" of this type is discriminatory. For the Fort Worth producer is not taking a lower profit on just the distant sales of temporarily surplus production, but consistently getting less profit from the ordinary customers on one side of his plant (the Austin-side) and more from those on the other (the away-from Austin-side) . Such, in crude outline, is the multiple basing-point system. We are by-passing many interesting and often unanswerable questions, such as: (1) are basing-point prices necessarily 10 In the recent Senate inquiry, " the :representatives of steel producers did not defend the charging of phantom freight, which had previously. been the custom in that industry" (Interim Report, p. 80). A 50 year practice (p. 2) can be wrong. THE MORALITY OF BASING-POINT PRICING 359 collusive? 20 (2) if they are collusive, who determines the basing-point areas and the basing-points? (3) what are the effects of basing-point pricing on new plant location? These questions are also beside the poinL The outline given above, together with a review of some actual cases, is sufficient for a moral judgment on the practice. Act'ual Cases There has been little complaint about basing-point systems in the past because, generally, a buyer of certain minerals o:r chemicals is only another producer, or perhaps a building contractor. These items are, generally, a small part of his total costs. H he should complain, no other producer would offer him a lower price; besides, his complaining would only risk his supply of such materials, especially in periods of short supply. Furthermore, the buyer knows that the other buyers in the area have to pay the same price; so they all stay quiet and pass the costs along to the consumer. Besides, producers can avoid outrageous injustices by establishing a few more basing-points, as happened in the early twenties when large increases in freight rates occurred. 21 Business men blundered badly, however, when this pricing practice was adopted by the glucose industry. 22 In 1945 the Supreme Court decided against the system in the Corn Products and Staley cases. The decision was not too difficult to make since the producers of glucose and dextrose established a single basing-point area for the whole United States. What occuiTed, 20 In the Conduit-pipe case, the FTC ruled that concurrent use of basing-point pricing by numerous competitors proves unlawful conspiracy (FTC Docket 445'l; 1948). Individual use of the system by a single firm, in the knowledge that it was being used by competitors, was also ruled illegal. Cf. Kaysen, C. "Basing Point Pricing and Public Policy," Quarterly Journal of Economics (Aug. 1949) LXII, 289. 21 In a masterly bit of prose, the U. S. Steel Exhibit No. 1418 (in the TNEC 1\'[onograph No. 42) remarks: " The increase in the number of basing-points then accelerated" (p. 42) . This is in line with the thesis that this pricing system ". . . had crystalized," and was " a nat mal result of basic economic conditions." It "has evolved." 22 Glucose is corn sugaro Dextrose is refined corn sugar. 360 RAYMOND C. JANCAUSKAS essentially, was a return of an old situation when steel products were sold throughout the United States at Pittsburgh prices plus freight. The Pittsburgh Plus system merited a cease-anddesist order from the FTC as early as 1924; now, twenty years later, the glucose industry comes along with a " Chicago plus " system. On top of that, glucose, unlike steel or cement, is not a fractional but a major part of its final product; it makes up 85% of maple syrup and up to 90% of cheap candy content. So it was to be expected that in this industry, where "the margin of profit of such candy manufacturers is so narrow that business may be controlled on a concession of one eighth of a cent a pound," 23 there would be trouble with an artificial pricing system. That is just what happened. The largest producer of glucose, the Corn Products Refining Company, was located at Chicago, the basing-point, but it also had a plant in Kansas City. Glucose was made at the Kansas City plant and sold to a local buyer for $2.49 per unit, with forty cents of the price covering a freight charge from Chicago to Kansas City. In another instance, the Kansas City branch actually paid 18¢ per unit to get it to Lincoln, Nebraska, but charged the 45¢ it would have cost had it come to Lincoln from Chicago. There were occasions when the fictional freight charge came to " approximately $400 per carload." 24 The price asked by the Kansas City plant might have been just if the costs at this plant ran about 40¢ higher than those of the Chicago Plant. Then the phantom-freight charge might be taken as a charge for the difference in production costs. 25 The defense did not choose this line of argument, and so 23 A. E. Staley Manufacturing Co. vc. FTC; FTC Docket 3803 in the U. S. p. ll. Circuit Court of Appeals, April Session 1944, No. ,. FTC Docket 3633 in the United States Court of Appeals, January Session 1944, No. 8116, p. 4. These amounts are "small pickings" when one considers that the amount of fictional freight for steel bars, shapes and sheets produced on the Pacific coast ranged from $10 to $13 a ton. Before July, 1938 some steel sheets were sold on the Pacific coast at " Pittsburgh Plus," with phantom freight p. lUi). amounting to $15 a ton. (TNEC Monograph No. 25 TNEC Mono. p. 36. THE MORALITY OF BASING-POINT PRICING 361 the basing-point system was ruled illegal by the Supreme Court precisely because of the discrimination, or injury to competition, that results from such pricing. The extra price paid by the Kansas City candy and syrup makers gave the Chicago producers of the same products a definite and systematic advantage in competition. For no reasonable cause, the Kansas City candy and syrup makers were being artificially discouraged from competing toward the Chicago area, because in buying their glucose they had to pay more on account of phantom freight. The closer their competitors were to Chicago, the less they had to pay for the corn products. So that if the fair price for glucose is $2.00 per unit in Chicago and, fairness aside, $2.40 per unit in Kansas City because the freight charge between the two cities is 40¢ per unit, every sale by the Kansas City producers towards Chicago supp_oses a raw material cost of glucose of, say, $2.35 or $2.10 according as the buyer is closer to Chicago. And it is simply impossible for Kansas City candy producers to sell in Chicago unless they meet the price of Chicago producers, after absorbing the freight from Kansas City to Chicago. Yet, at the same time, the Chicago producers had no difficulty in selling their products in Kansas City if they so wished, since their freight costs were part of the price of Kansas City producers. 26 Furthermore, if we suppose that the Kansas City plant is at least as efficient as the Chicago plant, with the costs identical in both places, and the freight charge is 40¢ again, then the Kansas City producers get 40¢ in windfall profits when they sell to Kansas City buyers. The Kansas City buyers get no benefit from being next door to their producers because the industry is pretending that the only source of glucose supply is at Chicago. As a result, for no reasonable cause, the Kansas City buyers are punished for setting themselves up near Kansas City glucose plants instead of in Chicago. In the last two cases we have supposed, several injustices •• Some candy makers actually moved to Chicago. Cf. Com Products Refining Co. vs. FTC, FTC Docket 8688 in the U.S. C. A., No. 8116, p. 4!. 362 RAYMOND C. JANCAUSKAS are apparent. The first injustice is the social (or " legal ") injustices of restricting markets of both candy and glucose producers by the basing-point mills. The candy producers outside of Chicago suffer in the manner described above; and since their business is restricted, it follows that their suppliers, the glucose producers outside of Chicago, also suffer. But, in neither case is there any violation of strict justice since no producer has a strict right to any sale. This is particularly clear when non-basing-point producers are partners to the scheme; and they generally are because the cramping of sales toward the basing-point is generally outweighed by the " phantom freight " that non-basing-point mills can collect on local sales. As an illustration of the latter point The President of the Laclede Steel Company of St. Louis testified that its plant had been located in St. Louis because of the prospect of making money " by selling for more than it cost, on account of the protection we got, on account of the Pittsburgh Plus in existence at that particular time." 27 One can readily see, then, why it would be ungracious for the Laclede Steel Company to complain about the smaller mill nets it had to take on sales toward Pittsburgh. However, if the non-basing-point mills are forced into this arrangement and their markets are consequently restricted artifically, those who force them into this arrangement are guilty of a sin against charity. 28 The basing-point mills can give no good reason why "' TNEC Mono. 42, p. 188. •• There are difficulties in classifying sins against social justice. The traditional division of " justice " is into " general " or " legal " justice and " particular " justice. Legal justice governs the relation of the individual to the community and exacts from the individual that to which the community has a right. Particular justice exacts from the community or from another individual that to which an individual has a right; in the former case it is called "distributive," in the latter "commutative." Howeve_r, difficulties arise in this classification because the term "community " is often taken as the government authority which distributes burdens and benefits. There is no place to put a group of buyers or producers who have certain rights that another group of producers must acknowledge. But if the term " community " is extended to refer to any person or group which distributes burdens or benefits whether to one or more persons, the classification has no gaps. Prolonged use of the wider concept leads to the realization that "legal justice " is only THE MORALITY OF BASING-POINT PRICING 868 they should cramp the markets of non-basing point mills. Both mills should have the same freedom of access to their markets in any direction. The golden rule is the touchstone of love of neighbor: would the basing-point producers be willing to change places with non-basing-point producers (supposing that adequate compensation is given to the producer who has the better plant) ? The only possible justification for charging " phantom freight " -might have been that Kansas City production of glucose was inadequate to meet Kansas City demand. Then, in order not to discriminate against those candy manufacturers who would have to get their glucose supplies from Chicago, the Kansas City glucose suppliers would charge all their customers the Chicago price. In other words, the economic rule that there can be only one price in one market has to be followed: " Phantom freight " charges, then, become a reward to the glucose producer in Kansas City for putting up a plant in a " deficiency " area. This reward is justifiable as an incentive to producers to increase capacity in Kansas City so that local supply may eventually equal local demand and get rid of the uneconomical transportations of glucose from Chicago to Kansas City. According to Prof. Machlup, the. economic phenomenon operating in this situation is not " pantom freight," but a price differential. 29 The seller is taking a price he may charge, under the pretense that the difference is due to freight costs. If this pretense were forbidden to him, he could get the same profit by merely establishing another basing-point by the previous freight cost. In the example given above, the Kansas City producers could have merely raised the ;Kansas City price by 40¢ per unit instead of charging " phantom freight " to that amount. The difficulty with this justification is that the companies must prove that there is only one market area, e. g. for glucose another way of looking at love of neighbor, specifically, what each business mail owes to his consumers or other producers. •• Op. cit., p. 150. 364 RAYMOND C. JANCAUSKAS in the United States. There must be statistical proof that all sections of the United States either continually or frequently have recourse to the main source of supply. If they do not have ·such recourse, it is obvious that, somehow or other, local demand is being satisfied by local supply. For example, if there are no glucose shipments from· Chicago to Kansas City and points west, we must conclude that another market area exists in the West. In that case the deficiency area argument does not apply at all. The " one market " argument applies in different degrees to various industries. It is a much stronger argument for products ·. like steel and cement since their supply depends on geological deposits; it is much weaker in the case of glucose since this involves the processing of an agricultural product. The strength of the argument also depends on the weight of the final product and on the transportation costs. The heavier the product, and the higher the transportation. costs, the greater is the tendency towards smaller market areas because of the high shipping and handling costs. The heavy transportation costs that are alleged to be the reason for a delivered price system are, oddly enough, the reason why such a system is hard to justify. 30 In summary, then, the defendants have never backed their references to the "deficiency" area argument with statistical proof. If they could have proven their case basing-point pricing would be morally justifiable. In that situation, there has to be one price at which demand and supply balance out throughout the market area. If in some comers of that area, or throughout the area in fact, the sales price does not seem to bear any close relation to costs, this is ·only because the fair price in •• The recent increases in transportation cost have been highly instrumental in persuading the steel producers to give up their fight for basing-point pricing. " Industry is starting to realize that it does not make much difference what the Federal Trade Commission or Congress does about f. o. b. mill prices. Freight rates gone so high, labor is so expensive, that it pays to exploit the area nearest the plant or plants." Thomas Campbell, "The Welfare State vs. Personal Initiative," The Iron Age (January 5, 1950), p. 126. THE MORALITY OF BASING-POINT PRICING 365 ordinary circumstances 81 must take account of demand .as well as supply considerations. In such circumstances the high price may be allowed as a means of rationing the product to those who are willing to pay the most for it. But if one corner of that area constitutes a self contained market, there is no reason for insisting that the one price system must cover the whole area. In a large country such as the United States, and when heavy costs are involved as in the case of certain metals, the proof that only one, or a few market areas exist, is a very difficult assignment. . Since market areas fluctuate in size, the stability. of an artificial· price system can also involve violations of social justice. A similar phenomenon, often called " inertia," occurs also in ordinary competitive-pricing, but it does not bother anyone's conscience. Someday someone realizes that supply and demand have been out of line, and the proper change is made with profit. But when some individuals take the responsibility of establishing a " fair price," and take " police action " against " price cutters," then price " inertia " is not an accident but a deliberate choice. For example, once a basing-point area is established, producers may be able to reduce costs of production without bothering to the base price. This would lead to high profits which, in a competitive situation, would entice new :firms or suggest the expansion of the old ones. But if entry of new firms can be blocked, and the production of the old kept at its previous levels, the individuals responsible for unwavering base-prices are setting an artificial monopoly price. In other words, the primary purpose of the pricing system (cf. p. 3), from which we have prescinded so far, may operate against the community of consumers an; for the oil industry, the "competitive price is at the "fringe." cr. note on p. ll. 38 Ibid., p. 147. This is from Mr. Wooden's letter to Senator O'Mahoney. 39 Ibid. These admissions were being made because, at that time, the industry was arguing that basing-point pricing was competitive because sellers were departing from the "right" prices (the industry had just passed through a period of low demand). 40 Since the law of the land recognizes only " competition " and " monopoly " some such term manufacturing must be done to make the legislators and lawyers aware of the intermediate cases. 41 New York Times, December H, 1948, p. 26. THE MORALITY OF BASING-POINT PRICING 369 monopoly very well, since the base price is allegedly competitive and the rest of the prices within the area are derived from that price. Accepting this supposition, then, that not a competitive but an administered price must be charged, one finds that the basing-point system is still discriminatory without cause. If the buyers from the base point pay a " competitive " price, and the buyers from non-basing-point mills pay an artificial price? The "logic" of the system involves "phantom freight" and restriction of markets no matter what the prices must be, competitive, workably competitive, or administered. So the solutions offered under the competitive supposition apply analogously to any other supposition. OBJECTIONS TO THE SoLUTION The last year witnessed a renewed drive to legalize the practice despite the Staley, Corn Products, Cement and Rigid Steel Conduit decisions.42 The drive was almost successful since, at the hearings before the Senate Subcommittee on Trade Policies,· the issue was confused, as is usual, and partly unavoidable, by discussions of monopolistic practices, zoning systems, conspiracies, the Clayton and the Robinson-Patman Acts, and much indignant defense of freight absorption for competitive purposes. The public found the matter -too complicated to get interested in it; and those who were interested in it got garbled or distorted accounts of the discussions, even in such a paper as the New York Times. 43 But, in the end, •• Prof. Fetter lists the numerous arguments that have been used by the industries since 19W in the American Economic Review for December, 1948. 43 A classic case of news distortion occurred in the December 11, 1948 issue, on p. 22, in an IU"ticle that "covered " the " Second 1948 Economic Institute" of the Chamber of Commerce of the United States, Dec. 9 and 10, 1948. One has to get a verbatim record of this Institute (available) to get the truth. One more: the National Small Businessmen's Association testified for basingpoint pricing before the Senate. Later it was discovered it was not a small businessmen's organization, but a "front " for other purposes, partly financed by U. S. Steel, Standard Oil and other companies. New York Times, February 22, 1950, p. 89. 370 RAYMOND C. JANCAUSKAS President Truman vetoed S. 1008 after considering "high economic policy and bread-and butter politics." 44 One of the last-ditch arguments of the basing-point protagonists is that f. o. b. pricing will establish local monopolies: The effect of required f. o. b. mill selling would be to give each supplier and his dealer a competitive advantage which in many instances might amount to a local monopoly in that area where he has a freight advantage. In areas where two or more competing dealers have the same outgoing freight advantage, the competitive advantage would accrue to the dealer whose product was purchased from the closest source of supply.45 In other words, a possible injustice is worse than the present actual injustices. Furthermore, the possible injuries are exaggerated .. It is only common sense that monopolies cannot charge any price they please; for the higher their price, the narrower their markets will be. In fact, by making its price higher than it was under the basing-point system, the local "monopoly" would destroy itself: The previous delivered prices as calculated under the basing-point system would become upper limits for the " monopoly prices " of the local mill, and the local mill can attain the distinction of being practically the sole supplier in its market territory only if it keeps its prices below those previously charged under the basing-point system. 46 Prices will, if anything, go down instead of up. The supposition is backed by the historical fact that, when the change occurred from £. o. b. pricing to basing-point pricing in the case of steel billets and pig iron (1933), prices increased by over $3.00 a ton. 47 As for the argument that companies with huge financial resources will drive others out of business with temporarily low prices, the answer is that the same holds true under basing"B. J. Masse, S. J., '"Muddle over Basing-point Prices," America (July 8, 1950), p. 871. ••Interim Report, p. 82. •• TNEC Mono. 42, p. 187. •• Machlup, op. cit., p. 104. THE MORALITY OF BASING-POINT PRICING 871 point systems. Price-wars have actually been going on, not in terms of price, but in terms of mill-net returns, out of sight of the public. That is, a plant at basing point A can invade the territory of basing.tpoint B, charging the prices that trade journals indicate for territory B. · But since the freight costs are actually much higher than those charged to the customer, the mill at basing-point A absorbs the difference, thus receiving a lower mill net from the customers in territory B than those in its own territory. In this way, by concentrating on sales in territory B, whi.le getting its ordinary prices in its own territory, the plant at basing-point A can ruin plant B. Plant B, of course, can reciprocate by stealing customers from territory A. If both are successful in stealing customers from each other, the volume of business remains the same but there is costly and wasteful "cross-hauling" of goods going on (particularly since transportation costs are generally high :for these industries). Obviously, freight absorption and taking business away from competitive basing-points can. continue as long as any company's finances hold out, or the larger volume of business pays for the lower profit on extra-territorial sales. So the Economic Research Department of the Chamber of Commerce of the United States asks: If, under basing-point pricing there are stopping points for reductions in mill-net returns, why would there not be substantially analogous stopping points for reduction in mill-net returns under ... f. o. b. pricing? 48 The only difference in the two situations is that, under basingpoint pricing, the buyers do not get the benefit of lower prices while the "war" is going on. 49 We may conclude, then, that even if f. o. b. pricing is restored (it is not the only alternative), small producers will not be driven out of business-for the same reason that they are suffered to exist today. 48 Chamber of Commerce of the United States, Delivered Pricing and the Law, p. 20. 49 There is some benefit when other producers become impatient and establish " cut-throat " price in the locality of the price-cutter. For an example, ct Edwards, C., op. cit., p. 831. 372 RAYMOND C. JANCAUSKAS Another argument for the basing-point system is that various industries have developed according to this price system. To abolish the system would be to ruin these investments. There would be great hardships in particular cases "probably without on balance improving the force and effect of competition." 50 As Prof. Machlup points out, there are two ways of understanding an" economic loss": the loss may be private or social. A fire that destroys a factory is both a private and a social loss. A fall in the price of inventory goods is a private, but not a social, loss. Losing exclusive rights to a patent is a private, but not a social loss. If competitors devise more efficient machinery, this js a private, but not a social, loss. Now, applying the argument to the present case, more efficient location of industries is in the same category as more efficient methods of production; the private losses of those who are inefficiently located would be on a par with the private losses of those who still follow antiquated methods of production. Consequently, . . . this " destruction of capital values," this "dissipation of investments in plant· facilities," would be (no cause) for regret by anybody who is concerned with the people as a whole rather than particular individuals, and with the national incoine rather than the private incomes of special groups." 51 EFFECTS OF BANNING BASING- POINT PRICING Corwin Edwards lists 52 the four most likely changes that will occur if basing-point systems are abandoned by all industries: (I) consumers will be given the full benefit of water and trucking rates where these forms of transportation are available and desirable; (2) there will be less "extravagant interpenetration of markets, involving excessive cross-hauling and other unnecessary expenses of sale and delivery, by depriving buyers of any incentive to purchase from nearby mills rather than distant mills .... "; (3) " ... there will no longer be an incentive for supplies and consumers of industrial products to ignore J. M. Clark, in the American Economic Review, XXXlll, 284. Machlup, op. cit., p. 259. •• Op. cit., pp. only the quoted parts are in Mr. Edward's words. 60 61 THE MORALITY OF BASING-POINT PRICING 873 . ."; some of the advantages of being near one another. (4) non-basing-point mills will get more business in the direction of the previous basing-points if the " competitive " price at the base point remains in force. It does seem, then, even apart from ridding the economy of discriminatory practices, the banning of basing-point pricing will bring improvements in over-all efficiency that should outbalance any private hardships. If there are proven damages to industry, the public might reimbuse the industry. Summary There are two necessary injustices in basing-point pricing, and a third that_may easily occur. The first necessary injustice lies in the artificial restriction of the markets of non-basingpoint mills. No restitution is due even when the damage is serious. The second necessary injustice, when there is no question of a" deficiency" area, is the charging of" phantom freight" by non-basing-point mills to customers so situated that the actual freight costs are less than those charged. Restitution is due to the amount of the " phantom freight." In cases where the actual freight costs are more than those charged, the seller is practicing freight absorption that he considers economically justifiable; no special moral problems are involved in these cases as long as such expedients are sporadic occurrences and on a small scale. On a large scale, or as a continuing policy, the practice would fall under the category of " dumping " practices which I hope to treat at another time. Restitution is due in those cases where railroad transportation is forced upon the buyer. The third possible injustice is the charging of exorbitant prices, a circumstance that may easily arise under artificial pricing systems. The treatment of this case would fall more properly under a consideration of monopoly pricing. RAYMOND University of Detroit, Detroit, Michigan. c. JANCAUSKAS, s. J. EUCLID AND ARISTOTLE HE TECHNICAL and philosophical interpretation of Greek mathematics is dominated by the opposite views of Plato and Aristotle, which have influenced even modern mathematical philosophy. Briefly, Plato teaches that mathematical objects are intermediate between the Ideas and the sensible world: as such, they reflect the eternal relations of their own Ideas are separate from the external world; they are discovered and not invented; they are expressed in assertorical and not in problematical propositions. On the other hand, Aristotle thinks also that mathematical objects are intermediate between Being and the sensible world: but they cannot be deduced from the analysis of Being and have no separate existence; indeed, they are abstracted from the sensible world, which accounts for their being applicable to it. Hence Plato accounts mathematically, and Aristotle logically, for the rational character of the world. These doctrines, which were widely known in the Greek world, have influenced most thinkers in one way or another. Is it possible at all to determine which conception is paramount in the work of Euclid the mathematician? The problem is made difficult by the absence of any direct information from Euclid himself, and also by the cosmopolitan and eclectic character of the intellectual climate of Alexandria where Euclid lived. While scholars grant that the author of the Elements has widely used in his work the Aristotelian theory of demonstration, most o£ them would insist that his ultimate vision was Platonic indeed. We believe, however, that the Euclidian systematization of mathematics is an application of the rational theories of both the Academy and the Lyceum; and that the Aristotelian trends in Euclid are stronger than any other. But 374 EUCLID AND ARISTOTLE 375 this debate must be prefaced with an account of the Alexandrian atmosphere and of the work Euclid left to posterity. THE CLIMATE OF ALEXANDRIA AND EucLm's Wmucs The conquest of Egypt by Alexander the Great in 332 B. C. had important consequences for the development of learning. While Athens still remained the seat of literary interests and speculative philosophy, Alexandria became the dynamic centre of scientific pursuits and of the technical arts. Owing to its geographical remoteness and to the political rivalry between Rome and Carthage, the Nile Delta was less open to an effective external pressure than Sicily or Greece itself. The death of Alexander in 323 B. C. and the wisdom and might of the Diadochs, gradually robbed Athens of its political preponderance. }-,ounded at a point where East meets West, Alexandria soon rose to the status of a great intellectual and commercial metropolis. With their broad cultural horizon widened the expeditions of the lVIacedonian conqueror, the Alexandrians came into contact with the esoteric as well as the empirical doctrines of the Persians, the Babylonians, the Assyrians, and the Phoenicians. Moreover they had at their very door the ancient and complex knowledge of the Egyptian priests who had been the first masters of the Greeks and had still many things to teach in spite of thei:r declining influence. The fusion of the Hellenic and of the Eastern civilizations made Alexandrian thought more cosmopolitan, but also more specialized and more mystical. But on the other hand, these circumstances also favoured frequent exchanges between Greece and Egypt. The great philosophical schools of Athens had already given their best to the world: Platonists, Peripatetics, Stoics, Epicureans, Sophists and Sceptics could cross over the Eastern Mediterranean and widen the circle of their disciples. Yet the attraction of so many different and contradictory doctrines was tempered by the mystical interests and practical minds of the Alex- 876 THOMAS GREENWOOD andrians themselves. It is obvious that all these mutual influ. ences gave a great stimulus to intellectual pursuits, which could not be ignored by the leaders of the country. Learning was greatly encouraged by the Greek rulers of Egypt, and especially by Ptolemy II Philadelphus and Arsinoe his wife. The Ptolemies were responsible for the foundation and maintenance of the Museum and the Library, which were housed in the royal citadel. In these two famous scientific institutions, eminent thinkers, scientists, engineers, alchemists and physicians were able to carry out research and experiments; while the wise discussed philosophy and religious speculations. They imparted their knowledge to the younger men who first flocked around them and then brought learning to distant cities and countries, without losing contact with their Alexandrian masters. During that period, science learned to stand on its own merits and to develop single-handed remarkable specialized inventions. Mathematics and· astronomy were graced with the discoveries of such famous men as Euclid, Archimedes, Apollonius, Aristarchus, Erastosthenes and Hipparchus. And if the natural sciences did not produce any worthy sucpessors to Aristotle, the experimental genius of the Alexandrians manifested itself in the more practical sciences of engineering, alchemy and medicine. Philosophy, however, turned its back on science and became more speculative, ethical and religious. It lost its universal character and encouraged objective compilations of past systems, instead of constructive syntheses based on the new scientific knowledge of the time. This situation did not seem to encourage the growth of a mathematical philosophy comparable to that of the Hellenic period. True, Neo-Pythagoreanism and Neo-Platonism have used numbers as an integral part of their structure. But the numerological aspect of these systems was less scientific than mystical in character. On the other hand, the connexion of the Alexandrian mathematicians with philosophy had not a metaphysical, but a marked methodological value. These trends are illustrated in the decisive work of Euclid the mathematician. EUCLID AND ARISTOTLE 377 The life and personality of Euclid have so far remained shrouded in mystery. The scanty information we possess about him comes from casual remarks from some later commentators. Most of the mathematicians who could have taught Euclid were pupils of Plato; a11d it is presumed that they gave him his scientific training in Athens at the Academy. It must not be forgotten, however, that the Lyceum also existed at that time; and that the influence of Aristotle was felt throughout the Greek world. So that if Euclid had been trained at the Academy at all, he must have been familiar with the current discussions of the day, with the textbooks used at the Lyceum as well as at the Academy, and especially with the Organon which gave to all thinkers a formal instrument of investigation and proof. Though practically nothing is known of the early life of Euclid, there is evidence that he flourished in Alexandria during the reign of the first Ptolemy (306-!'l83) because he is mentioned by Archimedes who was born just before the end of Ptolemy Soter's reign. He is even credited with the founding of a school in Alexandria where, according to Pappus, Apollonius of Perga afterwards " spent a long time with the .pupils of Euclid." 1 But such a school was unnecessary in Alexandria which possessed scientific institutions under royal patronage; and it is difficult to see, in these circumstances, how a mathematical school could survive Euclid by two generations. It is more reasonable to think that Euclid taught mathematics at the Museum in Alexandria, where he formed many disciples. This view justifies more readily the statement that Apollonius spent a long time in Alexandria with the pupils of Euclid; and the story of Euclid's reply to Ptolemy, that there is no royal road to geometry. An incident reported by Stobaeus shows the high conception Euclid had of mathematics. A pupil, who had just learned the first propositions of geometry, asked what he would get by that knowledge, whereupon Euclid bade his servant give the 1 Heath, Manual of Greek Mathematics, p. 203. 378 THOMAS GREENWOOD pupil a piece of money " since he must make gain out of what he learns." There is also the testimony of Pappus, who praised Euclid for his modesty and his fairness to other mathematicians, quotjng the case of Aristaeus to whom Euclid gave credit for his discoveries on conics without attempting to appropriate his methods. The Summary of Proclus contains practically all that we know about Euclid, though his remarks seem to be based upon inference rather than direct evidence. We are also told that Euclid wrote several works on pure and applied mathematics. Those which can be attributed to him definitely are the Elements, the Data, the book on Divisions, and two treatises: the Phenomena and the Optics dealing with applied mathematics. Many other works are unfortunately lost, among them the Porisms, the Plane Loci, the Conics, and the Pseudaria on fallacious solutions; while some others, which have come down to us under his name, are not his own compositions. As regards its mathematical content, the Euclidian Corpus offers few additions to previous knowledge, with the exception of some original proofs and some new theorems. Indeed, Euclid is praised less for his inventiveness, then for his synthetic genius, logical rigour and perfection of form. These qualities are brilliantly shown particularly in the Elements (o-Tmxe'ia), his principal work, which is considered the greatest elementary textboQk in geometry of all time, and a most notable witness of our debt to Greek mathematics. The technical importance of Euclidian synthesis is due to its factual completeness and methodological structure. Before Euclid mathematicians had obtained many remarkable results which were expounded either in special treatises covering specific problems, or in systematic collections as those of Hippocrates of Chios (c. 470 B. C.), Leon (c. 400 B. C.) and Theudius of Magnesia (c. 370 B. C.), which were used as textbooks. The special treatises give an orderly exposition of investigations relating to some definite mathematical problems. But the systematic collections of propositions involved the discovery of certain leading theorems bearing, in regard · to EUCLID AND ARISTOTLE 379 those which follow, the relation of a general principle by which many properties could be proved. Such theorems were called elements (crrotxeZa) as their function resembles that of an alphabet in relation to language. In this sense, it is said that Theudius "put together the elements admirably, making many partial propositions more general," and that Hermotimus of Colophon " discovered many of the elements," 2 as was also said of Euclid later. But the mathematical discoveries and methods of the fifth and fourth centuries had outgrown the rudimentary school texts of the time. Further progress required a well-organized inventory of the available material, including the Eudoxian theory of proportion. This was the task Euclid set himself to do: thus " in putting the Elements together he collected many theorems ofEudoxus, improved many propositions of Thaetetus, and gave an irrefragable demonstration of statements loosely proved by his predecessors." 3 These investigations required necessarily the alteration of the arrangement of the books in the earlier Elements, the redistribution of propositions between them, and the invention of new proofs applicable to the new order of exposition. This successful effort was considered so important that Euclid is still known as the " Author of the Elements " (8 as Archimedes was the first to call him. Indeed, it shows sufficiently Euclid's ingenuity and acumen, even though he left no record of more special research in mathematics, and made no claim to originality in his extant works. The test of Euclid's achievement would be to compare his work with proofs given by his predecessors. In the absence of any earlier manuals (for these must have been displaced by Euclid's own), one of the best sources is Aristotle himself. }'or his frequent mathematical illustrations imply that he had at hand some textbook containing most of the things he mentions, probably that of Theudius. By comparing corresponding state2 Proclus: Commentaries on the First Book of Euclid (ed. Friedlein, Leipzig, IS73), p. 67. 3 Ibid., p. 68. 380 THOMAS GREENWOOD ments in Aristotle and Euclid, one is able to follow the changes made by Euclid in the methods of his predecessors. For example, we find in the Aristotelian treatises the equivalents of Euclid's first six definitions, with the exception of that of the straight line for which Plato's definition is given. This may be taken as a fair indication that Euclid's correlative definitions of the straight line and plane were his own. Similar figures are defined identically by both, as is also the case with some of the terminology of proportions, for which Eudoxus no doubt provided the materiaL But Aristotle has certain terms like inflected and verging lines, which are not used by Euclid. He gives also some theorems which are not found in Euclid, like the one about the exterior angles of any polygon being together equal to four right angles, and like those about certain properties of the circle concerning plane loci and isoperimetry. In a striking passage 4 Aristotle points out that the theory of parallels involved a vicious circle. In that case, even if the leading theorems on parallels were known before, Euclid seems to be the first to solve this difficulty by formulating the famous postulate upon which he based his own system. The fact that Aristotle does not give any examples of geometrical or mechanical postulates leads one to presume that the classical postulates relating to the straight line, to the right angle, to the parallels, and to the construction of lines and circles, may have been explicitly established by Euclid himself. In spite of his logical theory of demonstration, Aristotle has not always given rigorous and final proofs of the mathematical propositions mentioned in his works. Many of them involve certain assumptions far more complex than the propositions to be proved. Some others differ from those of the Elements, as the one about the equality of the angles at the base of an isoceles triangle where 'inixed angles are used, 5 which is given as an illustration of the rule that the two premises of any syllogism must have between them an affirmative and a universal proposition. This is also the case with the proposition 4 Posterior Analytics, 76 b 9. 5 Ibid., 94 a 30. EUCLID A.J."'D ARISTOTLE 381 that the ·angle in a semi-circle must be a right-angle 6 which Aristotle proves in two stages. He shows first that it is true for an inscribed isosceles triangle with the diameter as its base; for the median of the given angle bisects it and forms two right-angled isosceles triangles; and the proof is completed by using' the equality of the angles in the same segment. The demonstration of Euclid (Bk. Ill, p. 31) is more direct and general. From the strict standpoint of methodology, there are technical deficiencies in the works of all pre-Euclidian mathematicians. In the case of Aristotle, however, one must bear in mind that he was not concerned with mathematical research proper, but rather with the formulation of his logical and physical theories, and with their illustration by means of, the best mathematical examples given in the current manuals of his time. In" putting together the elements," Euclid set himself a different task which he accomplished with a remarkable success. Hence, the Elements is not only a complete compendium of elementary Greek mathematics, but also an illustration of the powerful combination of the Platonic method of analytic regression and synthesis with the Aristotelian conditions of logical necessity and demonstration. Before Euclid, philosophers had discussed and set down the characteristics of science in general and of mathematics in particular. Plato and Aristotle had pondered over the meaning of the latter, mentioned or described its features and method, and justified it with ontological principles or epistemological considerations. In this task they proceeded either from the consideration of the rational conditions of science, or from the analysis of the results obtained by mathematicians proper. But neither Plato nor Aristotle were actually interested in improving the arrangement or the proofs of the elements. The former expounded his dialectical method and used mathematics for the benefit of his Theory of Ideas and of his cosmology; while the latter gave mathematical illustrations to make good his generalizations concerning the struc• PriM Analytics, 41 b 15. 3 382 THOMAS GREENWOOD . ture, principles and method of deduction and demonstration. Yet, both provided Euclid with the rational mea:qs of systematizing the exposition of geometry. However, there is no reference to Plato or to Aristotle in the Elements or in the Euclidian corpus for that matter: hence the controversies about Euclid's philosophical allegiance. A preliminary analysis of the Elements would to understand the problems and the solutions offered. I THE pATTERN OF THE ELEMENTS The presentation of the thirteen books of the Elements is direct and strict as befits a textbqok. Without any apologetic introduction or directional principles, the first book opens bluntly with twenty-three definitions relating to such fundamental concepts as point, line, surface, volume, circle, angle and figure. Without any comment, we are then given five postulates referring to the construction of straight lines and circles, and five original axioms or common notions which constitute the basis of geo;metrical reasoning. The difference between these types of statements is given by their content and not 'by any explanations. Among the postulates are the two properties of the straight line: (a) that two straight lines cannot enclose a space 7 and (b) that two straight lines in a plane will meet when produced, if a third lines cuts them so as to lorm on the same side two interior angles together less than two right angles. The postulate • The wording of the original definitions, postulates and axioms is not necessarily that of"Euclid in all the There hav'e been so many transcriptions and editions of the Elements, that scholars and copyists have tried some minor improvements here and there by rephrasing, substitution or insertions. From the earliest times, however, the Euclidian straight .line has been characterized by two postulates like those. .Euclid's sagacity in this matter is. illustrated by the failure of subsequent mathematicians to prove either of them. The failu;re of all such attempts led in the nineteenth century to the discovery of new systems of geometry __ characterized by different types of straight lines, which have caused endless philosophical and logical controversies. We have attempted ourselves to solve some of these difficulties with a new system of Euclidian axiomatics (Essais sur la Pensee Geometrique, 1948) involving a hypothetico-deductive rearrangement of the basic Euclidian intuitions. EUCLID AND ARISTOTLE 883 requiring the equality of all right angles is equivalent to the principle of the ,invariablity of figures, which makes congruence possible. The axioms are statements about the equality and inequality of magnitudes. These opening definitions and hypotheses are followed by propositions about triangles and the mutual relations of their component parts. It is interesting to note that the first proposition refers to the construction of an equilateral triangle. In these initial propositions are proved the properties of vertically opposite angles, adjacent angles, perpendiculars, and congruent triangles. The theory of parallels (pp. which requires postulate (b) given above, leads up to the theorem that the interior angles of a triangle are together equal to two right angles. We come then to the areas of parallelograms, triangles and: squares (pp. 33-48) , including some cases of the Pythagorean method of applying areas. The book ends with the famous proof (and its converse) of the relation between the square of the sides of a right-angled triangle and the square of its hypotenuse, a relation discovered by Phythagoras, according to tradition. The second book proceeds with the theory of the transformation of areas, and proves "!!he equality of sums of rectangles and sqpares to other such sums. It introduces the use of the gnomon for the solution of numerical problems; and it gives the initial theorems of the geometrical algebra with whicb the Greeks solved elementary algebraic equations by means of geometrical processes and proofs and wit.hout using algebra proper. Just as Book I ends up with the Pythagorean theorem of the square of the hypotenuse, so Book II leads up, with propositions and 13, to a generalization of the theorem for any triangle with sides a, b, c, proving what is equivalent to the modern formula a 2 = b2 c 2 - 2bc_ cos A by geometrical means. Books III and IV develop the geometry of the circle. It begins with definitions, such as those of equal circles, tangent, chord, segment, sector, similar segments, angle in a segment and the archaic notion of the ' angle of a segment ' referring to + 384 THOMAS GREENWOOD the mixed angle made by the circle with the chord at either end of the segment. Then it goes on to prove propositions dealing with the form of the circle, intersecting circles and tangent properties; and it finishes up with the beautiful demonstration of the constant value of the product of the two rectilinear segments OM. ON of a straight line, cutting a circle at any two points M and N, and passing through any point 0 internal or external. Book IV proceeds to the inscribed and circumscribed polygons constructible with straight line and circle, the most important being the regular pentagon sacred to the Pythagoreans and the regular fifteen-sided polygon used in astronomy. Books V and IV expound the general theory of proportion applied to commensurable and incommensurable magnitudes of any kind, according to the Eudoxian method. A ratio is considered as a sort of relation of size between two :6.llitemagnitudes of the same kind. Book V introduces infinite magnitudes, and defines quantities in the same ratio or in greater ratio, and the transformation of ratios by alternation, inversion, composition, separation and conversion. It proceeds to numerical multiples and equimultiples; and it proves the validity of the transformation of one proportion into another. In applying in Book VI this general theory of proportion to plane geometry, Euclid proves first the fundamental propositions that two sides of a triangle cut by a third side are divided proportionally. Then he shows its various consequences in the construction of proportionals and in the similarity of triangles; and finally, he uses the Pythagorean theory of application of areas in its most general form, with results which are equivalent to the geometrical solution of a quadratic equation having a real and positive root. The book ends (VI. 31) with a remarkable generalization of the theorem of the square of the hypotenuse, showing that proposition to be true not only of squares, but of three similar plane figures described upon the three sides of the right-angled triangle and similarly situated with reference to the sides. Books VII, VIII and IX deal mainly with the nature and EUCLID Al\TD ARISTOTLE 385 properties of rational numbers represented throughout by straight lines and not by numerical signs. Following the traditional conception of the Pythagoreans, Book VII begins with the definitions of unit, number the varieties of number, including plane, solid and perfect numbers; then it demonstrates some elementary properties and operations referring to various kinds of numbers. Books VIII and IX relate to numbers in continued proportion (geometrical progression); and the latter proves that a number can only be resolved into prime factors in one way, that the class of prime numbers is infinite, and other important propositions. Book X, the most finished of the whole work, deals with irrationals understood as straight lines which are incommensurable with any straight line assumed as rational. It begins with the famous proposition on continuity which is used by the Method of Exhaustion, and which was completed by Archimedes and known under his name. It states that if from any magnitude there be substracted its half or more, from the remainder again its half or more and so on continually, there will remain a magnitude less than any given magnitude of the same kind. The whole subject was originated and expanded by Theaetetus. What we owe to Euclid in this matter, according to Pappus, is the precise definition, classification and exposition of rational and irrational magnitudes. The elaborate array of definitions and proofs given in this Book is due no doubt to the absence of the conception and symbolism of algebra. For the Greeks had to represent with straight lines operations which deal with the solution of equations and the discussion of their roots. But as all straight lines look alike, a detailed classification of linear definitions was necessary to cover all the operational distinctions required by the subject. The methodological contributions of Euclid to the systematic treatment of irrationals appear to be more numerous and elaborate than usual; for they will be used in Book XIII for the complete determination of the regular polyhedra. Finally, Books XI, XU and XIII deal with solid geometry 386 THOMAS GREENWOOD which appears less systematized, however, than plane geometry. In some proofs Euclid allows more abrupt leaps than he permits himself in the earlier books; and the distinction between congruence and symmetry is not always clear. But this was, of course, the first attempt to organize solid geometry into an exact science. The required definitions are given in Book XI where the order of propositions is very similar to that of the first books. After proving a series of properties of straight lines and planes in space, it deals with parallelipipedal solids. The Method of Exhaustion is the core of Book XII and serves to determine areas of circles and volumes of the solids as well as various proportions between their elements. It is well known that the Method of Exhaustion was invented by Eudoxus (c. 408-355 B. C.) in answer to Zeno's dilemmas about the infinitely small: it showed that the mathematician does not require actually such an infinite, but only the possibility of arriving at a magnitude as small as we please by continual division. This is the Greek version of the modern method of limits, which allowed the mathematician to evaluate a magnitude by calculating others close to it by defect or excess and eliminating by a reductio ad absu,rdum the unwanted alternatives. Lastly, Book XIII deals with the construction of the five regular solids (tetrahedron, oCtahedron, cube, icosahedron and dodecahedron) and the determination of a circumscribing sphere. A number of preliminary propositions have to be proved before constructing the sides and angles of the polyhedra and determining the relations of those sides with the radius of the circumscribed sphere. It remains to say a few words about the other works of Euclid. The Data developed some details of the subject-matter of Books I-VI of the Elements, with special reference to the construction of plane figures by means of some given elements. An example of the alternative methods used here is the solution of the simultaneous equations y-+- x =a and xy = b2 , which is another form of the solution of the quadratic equation ax -+- X 2 ---..: b2 given in the Elements (II. 5, 6) . The initial definitions EUCLID AND ARISTOTLE 387 of the various meanings of the word given proposed by Euclid in the Data are among the interesting features of this work; straight lines, angles, areas and ratios are given in magnitude when we can find others equal to them" Rectilineal figures are given in species when their angles are severally known and also the ratios of the sides to one another" Points, lines and angles are given in position when they always occupy the same place" But the main purpose of the Data is obviously to help in shortening the analytical processes which are preliminary to a problem or proof: when we know that certain elements of a figure are given and that other parts or relations are also given by implication, it is often superfluous to determine that figure by an actual operation" The book On Divisions corresponds to the description of the original work given by Proclus in his Commentary" The general purpose of the propositions it gives in strict logical order, but often without proof, is the division of plane figures by transversals or parallels into parts having equal or proportional areaso This treatise has been edited in 1915 by R Co Archibald on the basis of the original text in Arabic discovered in 1851 by Woepke, and the portion of Fibonacci's Practica Geometriae dealing with the division of figures, which is supposed to have been written with the help of Euclid's work Of the two treatises dealing with applied mathematics, the Phenomena develops the geometry of the sphere according to the requirements of observational geometry" It contains the definition of the horizon, which is given for the first time as a single technical word: " Let the name horizon be given to the plane through us passing through the universe and separating off the hemisphere visible above the earth." The Optics deals with problems of perspective, explaining how figures look from different points of view or at different distances, as compared with what they areo Heath believes that this book may have been intended as " a corrective of heterodox ideas such as those of the Epicureans who maintained that the heavenly bodies are of the size they look," 8 Like the Elements, the Optics opens 8 Heath, op. cit., p. fl67" 388 THOMAS GREENWOOD with a series of definitions concerning the fundamental concepts of light and vision; and it proves a succession of propositions in strict deductive order. Indeed, all the works of Euclid reveal a methodological preoccupation which is paramount, and which offers much scope to the philosophical critic, both as regards its actual technique and its metaphysical implications. THE EucLIDIAN METHOD AND ITS IMPLICATIONS As a mathematician, Euclid is careful to remain within the definite realm of geometry. Perhaps the thought never occurred to him to venture philosophical pronouncements on matters already specialized as a science. In fact, his works make no allusions to Plato or to Aristotle, not even to their strictly methodological theories. Yet their philosophical implications are enshrined in his very method of presentation, and particularly in the exposition of the Elements to which we shall restrict our interpretative remarks. The Euclidian method of i1·rejragabledemonstration, as Proclus calls it, appears to be an elaboration of the principles which both Analytics of Aristotle discuss at length. In the Prior Analytics, we are told that no proposition should be admitted without showing its logical connection with earlier propositions already granted. It is true that Plato had also mentioned this rule of rational necessity; but his purpose was to force the mind back through such an analytical regression to the ultimate principles embedded in the Ideas. But Aristotle and Euclid with him meant only to base the whole deductive process on a small number of first principles stated at the beginning of a science, and granted as such without further regression. Going further than his master, Aristotle specifies the type of such primitive elements in the Posterior Analytics 9 where he mentions the definitions (opot), the postulates lvvotat) or axioms. (alr7Jftara) and the common notions Now, Euclid follows exactly this pattern, thus giving in the Elements the earliest evidence of a systematic arrangement of • Posterior Analytics, 74 b 5-'n a 30. EUCLil) 889 AND ARISTOTLE geometry beginning with these assumed elements, which are actually given as the basis of all subsequent deduction. Without enunciating all these Aristotelian principles explicitly, or even trying to state in words their distinctive characters, Euclid begins abruptly the first book of the Elements with his famous definition of a point as that which has no part. His actual geometrical statements, however, show conclusively that he was familiar with the Aristotelian requirements: in other words, Euclid does technically what Aristotle proposes theoretically. This is particularly true of the definitions, which Euclid merely gives without discussion and whenever needed, usually at the beginning of a theory. This practice indicates that Euclid shared the Aristotelian view 10 that mathematical definitions are separated from existence. As such they assert nothing as to the existence or non-existence of the thing defined: being simple answers to the question what is a given thing, they do not have to say that such a thing exists. Indeed, Aristotle maintains 11 that existence is neither an essence, nor a genus, nor a quality. In order to be justified, the existence of a thing must be assu:tned or proved. It is assumed, when we have a clear intuition of that object. 12 It is proved, when 13 we show not only what a thing is (T£ ecrT£v) but also why it is (S,a T;, ecrT,v) by means of a construction. This is particularly true in geometry where, as .Aristotle says, only points and lines must be assumed to exist; while the other notions must be proved to exist through some specific additions to the mechanism of proof. But this is precisely the Euclidian standpoint: in the Elements, the possibility of the definitions is neither questioned nor postulated. The definitions are merely asserted or proved, which entails that Euclid does not consider the objects they stand for as mere inventions· of the mind, pointing to a strictly nominalistic or even pragmatic hypothetico-deductive conception of geometry. The question arises here as to the type of intuition Euclid had of the geometrical objects he defines. If he were a Platonist, 10 Ibid., 92 b 10. Ibid., 92 b 14. 11 12 Ibid., 76 b sq. 18 Ibid., 90 a 81. 390 THOMAS GREENWOOD he must have intuited them as the reflections of their ideal and eternal paradigms. But then, he would have failed on the count of methodology: for as a Platonist, he should have carried the process of analytic regression beyond the very notions he takes as ultimate in order to discover how points and lines and other such figures can be accounted for by means of the One and the Dyad, which are the most primitive elements of the system. Yet Euclid, who has so much to say about numbers in the later books of the Elements, does not choose to go beyond points and lines at the beginning of his geometrical exposition. In doing so, he rather favors the Aristotelian view that such notions are the products of abstraction as applied to bodies of our ordinary experience. It is not the contemplation of the Ideas which gave Euclid the basic notions of geometry, but ,the usual data of the sensible world: and these make possible both their apprehension by successive abstraction and their application to the physical world. To be sure, even the biological categories of the Stagirite are hinted in Euclid's definitions, because these entail physical intuitions or experiments, because they involve a distinction between genus and species, and because they assume deliber. ately nominalistic flavor in order to avoid a confusion between the physical and the ontological orders of being. The use of the class-concept, which is Aristotelian, is indeed more than the use of the relation-concept, which is Platonic, in the EuClidian definitions. It is true that many of Plato's definitions of geomeJ:-ricalnotions have also an experimental origin. But we must not forget that experience comes after the contemplation of Ideas; while for Aristotle experience comes before the conception of abstract notions. And nothing in Euclid's Elements encourages us to believe that the various geometrical objects he asserts, constructs or analyses, are reflections of similar but substantial Ideas. With regard to the definitions the existence of which is proved, Euclid again follows the Aristotelian directives. The assimilation of the essence of a thing with its formal cause 14 a "Ibid. EUCLID AND ARISTOTLE 391 accounts for the transition from the subjective assumption of a basic geometrical notion to its objective proof. For the constructive process involved in such a proof postulates the Aristotelian principle 15 that in order to know what is an object or a thing, we have to know that it is. In other words, the definition of a notion is not complete until it is made genetic; for it is the producing cause which first reveals the essence of the notion. · Where existence is proved by construction, the cause and the effect appear together. 10 It is in accordance with these views and with the practice of earlier mathematicians that Euclid assumes the possibility of constructing straight lines and circles in the first three postulates. The other notions are defined and afterwards constructed, as for example the equilateral triangle (Bk. I, Df. 20 and Pp. 1), the right angle (Bk. I, Df. 10 and Pp. 11), the square (Bk. I, Df. 22 and Pp. 46), and the parallels (Bk. I, Df. 23 and Pps. 27-29). The difficulty of constructing all geometrical notions with the original Euclidian assumptions is not discussed by the author of the Elements, though it was known to earlier and especially to later mathematicians. As regards the characteristics or conditions of a correct definition, we are told by Aristotle that: (1) the different attributes of a definition, taken together, must cover exactly the notion defined; and (2) the different attributes of a definition, taken separately, must refer to the things better known or logically prior to the notion defined.U Thus Euclid defines a square (Bk. I, Df. 22) by means of the notions of figure, foursided, equilateral, and right-angled, each of which is wider, prior or better known than the term defined,· and which cover exactly the notion of the square when taken together. Definitions breaking either of these rules are There are several ways of breaking them; as for example, when a notion is defined by its opposite, or a coordinate species, or a synonym, all of which are coextensive with the notion to be defined. 15 Ibid., 98 a !tO. 16 Cf. Trendelenburg, Erlaiiterungen, p. 116. Topics, VI, 4-141 a !t6 sq. 17 THOMAS GREENWOOD There are several examples of such definitions in Euclid, among them the straight line "lying evenly with the points on itself": the expression " lying evenly " can be understood only with the very notion to be definedo It is interesting to note that Euclid never states a primitive notion without defining it; though such a practice is current today as a requirement of logical rigor Hence he does not merely assume points, lines and surfaces; but he also defines them by means of notions possibly satisfying our intuition without being justified in the system itselt In doing so, however, Euclid improved apparently the definitions of his precedessorso Indeed, Aristotle himself had criticized the earlier definitions of a point as the extremity of a line, a line as the extremity of a surface, and a surface as the extremity of a solid, by saying that they all define the prior by means of the posterioL 18 Nevertheless, Euclid must have felt that his definitions of these and of some other notions did not fulfil all the requirements of a correct definition as laid down by Aristotk he thought of supplementing them with the very statements Aristotle had critizedo Hence we are told (Bk I, Dt 3) that the extremities of a surface are lines, and that (Bk XI, Dt 20) the extremities of a solid are surface so Yet, in spite of their technical shortcomings, these supplementary explanations help to form a better understanding of the Euclidian definitions, for we must consider first a solid, which is more closely related to the bodies of our experience, in order to understand correctly the definition of a solid, a surface, a line and a point, as results of successive abstractions" Similar remarks are suggested by most of the Euclidian definitions; and not only by those of such controversial notions as the straight line and the. parallels, but also by the elementary geometrical notjons involving a construction. Without discussing these definitions as such in particular, a strictly technical task, we shall insist on the question of geometrical constructions, as they introduce the consideration of the hypotheses of 0 18 Ibid., Hl b !'lO. 393 EUCLID AND ARISTOTLE geometry, and as they involve a major distinction between the Plantonic and the Aristotelian conceptions of science. We are 'told by Plato and Aristotle that geometrical constructions are made logically possible and mathematically useful by the hypotheses of geometry which are of two kinds: the postulates and the axioms. 19 The first are more particular to geometry, as they beg certain properties of the straight line especially, on the ground of some idealized intuition or generalization no doubt, capable of justifying subsequent constructions and logical deductions. The second assert certain general properties of magnitudes, applicable to relations of equality and inequality between figures, and illustrating the elements of necessity in the actual process of demonstration. Euclid gives a set of propositions for each kind, but without explaining their logical character which he simply puts into use. Yet the classification of these statements shows that he knew the Aristotelian distinction between axioms which are KaO' EaVTa Or self-evident, and postulates which are accidental and not avayKTJ'> or necessary. Furthermore, the remarks of Proclus about axioms and postulates indicate that the Greeks were conversant with the logical and practical technicalities involved in their distinction. But certainly Euclid meant to get on with his mathematical exposition, rather than getting involved in discussions. Yet, neither the meaning nor the use of the geometrical hypotheses can be strictly independent from philosophical implications. To begin with, the argument already mentioned concerning the Euclidian definitions could be repeated here: If the geometrical hypotheses reflect properties of the Ideas, a e' 19 The term axiom, which is currently used by Aristotle, seems to have been introduced by the Pythagoreans; while the term common notions seems to be due to Democritus (Cf. Sextus ap. Diels, A. 111) who is also credited with a work on the Elements, the subject-matter of which is suggested by titles given by Thrasyllus (Cf. Diels, B. 11, n-p). In his Commentary on Euclid (p. 194), Proclus alludes to the habit of later mathematicians calling common notions what Aristotle defined as axioms, these terms being henceforth equivalent. However, Aristotle also refers to the axioms as T.t Kotva. or more rarely as Kotva.l llofa.t or common opinions (Cf. Metaphysics 996 b 25-30, and 997 b 20) implying that such statements are urged as requirements of common sense. 394 THOMAS GREENWOOD true Platonist should have tried to justify them by analytic regression in terms of more ultimate Platonic principles. Furthermore, if postulates express true properties of the ideal figures to which they refer, they should be stated assertorically. But Euclid proposes them as requests, that is, as something which is not intuitively clear and necessary. The various attempts to prove some of the postulates reveal their ontological weakness and their strictly hypothetical character. Finally, postulates are used mainly to solve problems and to justify constructions: but a problem involves a request again and a construction involves change, while eternal truths are changeless and assertoricaL If Euclid were a Platonist, he would have given a different expression to his postulates, and he would have avoided the distinction between theorems and problems which is a fundamental feature of his method, in order to make aU his statements assertorical. The weight of these difficulties is such that Platonists have tried to explain them away with psychological arguments. 20 The fact remains however, that Euclid has not built the foundations of the Elements according to a thorough Platonic pattern. To be sure, the Euclidian method follows a middle course between the Platonic conceptions as expressed by Speusippus and the pragmatic suggestions of the disciples of Menechmus the inventor of the conics. Speusippus proposed that all mathematical truths should be expressed as theorems, insofar as they reflect unchangeable and uncreated relations; while the practical mathematicians who followed Menechmus maintained that all mathematical statements should be considered as problems, insofar as they deal with constructions and with analyses of mathematical objects. But Euclid thought rightly that some mathematical propositi!)ns are straight deductions from previous ones, while others involve constructions: in choosing to distinguish between them as he did, he follows the Aristotelian conceptions which insist not only on the psychological priority 20 Cf. Abel Rey, L' Apogee de la Science Technique Grecque (Paris, 1948), pp. HiS-194. EUCLID AND ARISTOTLE 395 of experience, but also on the existential priority of individual objects. Indeed, the hypothetical character of the postulates and the zetetic expression of the problems entail an Aristotelian vision of the mathematical truths they assert, insofar as they point to an idealization of sense experience, rather than to a repetition of preconceived necessary relations or realities. However, these ontological and epistemological implications of the presentation of the Elements do not affect their strictly logical structure; for both Plato and Aristotle insist on the rational necessity of the deductive steps in mathematical reasoning. So that Euclid need not be considered as an Aristotelian simply because he uses basically the Organon in his work. It is interesting to note, however, the structural refinements Euclid added to syllogistic practice, in order to strengthen the binding character of a systematic deduction in the particular field of mathematics. Here, the propositions expressing theorems, lemmas, corollaries and problems 21 are connected in such a way that the elements or steps necessary to the proof of any one of them are legitimately given in the preceding propositions, so that one can reason backwards, so to speak, until the basic apriori data are reached. This analytic regression is made rigorous by the technical elimination of intuition in the detailed expression of each connecting link within and between the statements which make up in a body the science of geometry. The Euclidian pattern of demonstration requires three fundamental steps: the enunciation, the proof and the conclusion. But these are expanded into six for greater clarity: (1) the pmtasis or enunciation of the proposition in general terms; (2) the ecthesis or specification of the partic:ular data indicated by leters on which the demonstration will be developed; (3) the 21 The meaning of theorems and problems has been discussed. A lemma is an auxiliary proposition which is required in a demonstration without being essential in the general exposition of a theory. A corollary is the statement of a consequence of a particular demonstratiOn which is not necessary in dealing with subsequent propositions. The functional distinction of these four types of statements suggests a view of geometry which is more pragmatic and methodological than Platonic i.n character. 396 THOMAS GREENWOOD diorismos or statement of the conditions of possibility of what is required to prove or do in terms of the particular data; this is sometimes followed by a discussion of the limits of the proof; (4) the kataskeve or construction of additional elements to the original figure, which may be needed in the demonstration; (5) the apodeixis or proof, which draws the truth of the enunciation from the various data given or constructed, with the help of previous propositions, hypotheses and definitions; (6) the symperasm,a or conclusion affirming that the original statement satifies the conditions of proof. The rational process of demonstration is direct by synthesis or analysis, and indirect by reduction or exhaustion. Synthetic reasoning explicitates the pattern of demonstration just outlined: it actually achieves the successive logical connection of the mathematical truths, and thus corresponds to the objective and permanent organization of mathematics; whereas the psyare different. As such, the chological processes of synthetic method appears to be the most direct, not only insofar as it allows the conclusion to be drawn immediately from the data and the eventual constructions, bu:t also because it exhibits the strict rational order of truths which is the goal of all knowledge. This should not mean, however, that these truths reflect a world of substantial ideas; or at least that Euclid had this in mind in choosing a synthetic pattern as its principal method of exposition. For the synthetic process is just as effective and fundamental in the hypothetico-deductive vision of mathematics, which corresponds more closely to that of Euclid and to the Aristotelian tradition. The analyticproof is in a way the inverse of the former. It is a partial application of the general process of analysis, which allows the assertion of the truth of a statement by a logical regression to the fundamental definitions and hypotheses. It develops into two steps: the first is the apagogy or transformation, which assumes the truth of the proposed theorem or problem, analyzes the particular conditions of its proof, and then shows that these conditions can be provided legitimately for a EUCLID AND ARISTOTLE 897 direct demonstration of the original statement. The second step is the resolution which discusses whether the conditions of possibility (or diorisms) of the transformed proposition are together sufficient to carry assent. In short, the analytic proof actually transforms a given proposition into a simpler one, shows the original data to be sufficient for the proof of the simpler proposition, and determines that the proof of the simpler necessarily entails the proof of the original proposition. Indirect proof is either by reductio ad absurdum or by method of exhaustion. The former process the impossi-:bility of a proposition contradicting the original one, ·and concludes that the latter is true on account of the principle of contradiction. Such an indirect method is a kind of elaboration of the apagogic reasoning which is also a process of reduction. On the other hand, the method of exhaustion reduces proofs which lead to in:finitesimals to problems involving formal logic only. Thus it proves that an assumed relation of magnitude must be what it asserts, by showing that both assumptions of it being greater or smaller lead to absurdities. Here again, this type of demonstration appears in a way as a more complex elaboration of both the apagogic reasoning and reductio ad absurdum. As a rule, the indirect methods and more particularly the apagogic reasoning always follow immediately the ect.hesis in the general process of proof, when they are used, of course. The disadvantage of these indirect demonstrations is that they assume the result which has to be proved, so that mathematicians have to :find it first by other and more tentative methods. Such demonstrations are used when a direct proof is difficult or impossible, and not because Euclid " had to convince obstinate Sophists who plumed themselves on their refusal to accept more obvious truths." 22 Here again, Euclid's objective •• Clairaut, Elements de Geometrie (Paris, 1741), pref. pp. 10-11. A similar suggestion is made by Houel in his Easai critique aur les principes fondamentaux de la geometrie (Paris, 1867), p. 7 where he says: " Euclid's method is due to his desire to shut the mouths of the Sophists; hence his habit of always proving that a thing cannot be instead of proving it to be." 4 398 THOMAS GREENWOOD was to get on with the unrolling of his successive deductions according to the canons of logic, and not to skirmish with schools of philosophers about the meaning and possibility of truth, It remains to add that the general organization of the Elements and the actual order of succession of its propositions have been influenced considerably by the historical evolution of geometry. Traditionally and psychologically, the easiest but not necessarily the simplest generalizations or relations were discovered firsL And whenever some particular propositions were thus found, they were integrated into the group of truths of the same kind already known, Hence, it was natural for Euclid to have this in mind in writing his work, even though geometrical truths were not discovered in the logical order in which the Elements present them. For, indeed, it is logically possible to arrange the content of this work in many other ways: for example, one may begin with the arithmetical books, or even with more general principles; or again, one might put together the elementary and the general propositions referring to the same theory, which Euclid placed in different books, as in the case of the propositions on similarity. Thus it appears that in combining into a single synthesis the various geometrical theories known at his time, Euclid took into account what was done previously, but without sacrificing the demands of a deductive exposition. The success of Euclid's effort is proved by the fact that his collection and arrangement of the Elements have survived centuries of controversies, Later mathematicians were able to supply minor changes in their wording or disposition; but none was fundamental or final. This is the case, for example, of the theory of parallels which had caused such a storm in the history of mathematics. In as in any other theory, an analysis of Euclid's presentation shows that nothing is admitted that could be dispensed with, and that little that matters is left out. To be sure, there are a number of imperfections in Euclid's work: thus, a number of his definitions are intuitive, EUCLID AND ARISTOTLE 399 as are also some of his demonstratious; some capital hypotheses are omitted, as those concerning order and direction, and the postulate of indeformability of figures; again, logical proofs are often and needlessly preferred to arithmetical or geometrical demonstrations; and finally, systematization of geometry covers only a part of the field. Yet, the modern conditions of methodologicar rigor, and the various discoveries in synthetic and metrical geometry do not dim the brilliance of Euclid's work: are legitimate extensions based on different or generalized systems of axioms, or required by the rational systematization of the new material provided by the growth of mathematics. THE PHILOSOPHICAL VISION OF EucLID A mathematician can restrict himself to the study of his field, without having to choose of necessity aphilosophy to account for his labors. It is possible that Euclid would be content with his title of a-TotxewJT'Tj'>, especially as his works offer no direct information about his philosophical interests. But there is little doubt that he was acquainted with the speculations of his contemporaries, not only because of the intellectual climate of his age, but also on account of his very mathematical :training. Though specialization was more pronounced then, a division of labor did not preclude an interpenetration of the various disciplines, especialJy in the schools were the influence of the masters was paramount. On the other hand, if Euclid's mathematical teachers were pupils of the Academy, it does not follow that his technical training carried with it an acceptance of the Platonic doctrines. The suggestion that Euclid favored Platonism is made by Proclus for the reason that he "set before himself, as the end of the whole ElementS, the construction of the so-called Platonic figures." 23 But Proclus is careful to draw a distinction between the stridtly aim of the Elements and the Produs, Commentries (op. cit.), p. 68, 400 THOMAS GREENWOOD ultimate intentions of their author. The latter "is concerned with the cosmic figures"; while the former is bent on" making perfect the understanding of the learner in regard to the whole of geometry." 24 Heath finds fault with this distinction, because the planimetrical and the arithmetical portions of the Elements have no direct relation to the construction of the five regular solids as such. 25 Yet he admits casually that "Euclid was a Platonist " 26 when he shows that the Euclidian definition of the straight line is based on the definition Plato gave of that entity. And, in any cause, the fact remains that the Elements does end with the construction of the solids, in spite of their indirect relation to its earlier parts .. These circumstances agree much less with a rigorous exposition of geometry than with the alleged Platonic interests of Euclid, so that the initial statement of Proclus may be taken as something more than a mere attempt to connect the Alexandrian with Platonic philosophy. This idealistic interpretation· of Euclid is also maintained by recent historians 27 with more potent arguments. They try to show that the who1e structure of the Elements is a practical realization of the Platonic conception of science in general and of mathematics in particular. The discussion already presented of the imposing structure of the Elements has already shown, however, that Euclid's systematization of geometry is not merely an elaboration of the Aristotelian syllogistic and theory of proof. To be sure, it entails a number of considerations which are closer to the Aristotelian than to the Platonic world-view. As a special application of Aristotle's logic to the exposition (but not the invention) of geometry, it suggests that Euclid organized the geometrical elements in such a way as to encase them neatly in Aristotle's Analytics where the logic of classes predominates. •• Ibid., pp. 70-71. •• Heath, The Thirteen Books of Euclid's Elements (1926), Vol. I, p. !!. •• Ibid., p. 168. •• Cf. A. Rey, op. cit. But, on the other hand, L. Brunschvicg in Les Etapes de la Philosophie Mathematique, p. 89, considers the Elennents "as a veritable Analytics of geometry parallel to the Analytics of formal logic." EUCLID AND ARIS'fOTLE 401 Indeed, the biological conceptions of the Stagirite are hinted in Euclid's definitions, insofar as they involve physical experiments, as they use the distinction between genus and species, and as they assume a nominalistic flavor in order to avoid a confusion of the orders or levels of abstraction. The primacy of the class-concept asserts itself the more, when one remarks in the Elements the formal neglect of relational arguments, of spatial postulates, and of constructibility conditions, which are technically discussed by modern axiomatics. It is true that Euclid uses also factually a relational logic in his processes of construction and proof; for this is a natural condition of the activity of the mind and of the development of mathematics proper. But the logic of relations was not formalized before the nineteenth century, though Aristotle speaks of conditional arguments and though the Stoic had developed a practical theory of the hypothetical syllogism. Hence Euclid does not share Plato's anxiety for the status and use of relations as such. He is quite satisfied with the more positive organization of the geometrical elements on the broad basis supplied by intuitive concepts copied on idealized experiments or constructed logically with them as material. Such an Aristotelian outlook prevailed also in the systematization of the higher geometrical theories developed by Euclid's successors: instead of stressing the relational structure of these new fields, they described and organized them with the help of categories or classes more pliable to an analytical and a syllogistic treatment. The actual disposition of the mathematical theories which make up the body of the Elements provides a further argument against the alleged Platonic vision of Euclid. His work begins neither with number, the most immediate reflection of the Number-Forms, nor with the elementary triangles which Plato uses for the construction of the material world. For his purpose Euclid takes concepts which Plato would consider as intermediate between number and the elementary-triangles, but which Aristotle would obtain more directly from the external 1'liOMAS GREENWOOD world by intensive abstraction. Expanding this realistic approach, Euclid devotes the four first books of his work to a simplified systematisation of what may be called a natural geometry. He mtroduces much later the theory of irrationals, which a Platonist would have given at the very beginning. What is most significant, he the theory of irrationals on geometric considerations, which is not the Platonic scheme of accounting for geometry by means of numbers. In other words, Euclid geometrizes the continuum, instead of fulfilling the Platonic dream of its arithmetization. This factual priority of geometry cannot be explained away on strictly methodological grounds: for it points obviously to a vision which does not fit with the overall Platonic synthesis. Moreover, the loose connection in the Elements between planimetry and stereometry suggests that Euclid incorporated in them the Pythagorean theory of the regular solids for the sake of mathematical completeness rather than as an illustration of his alleged Platonic faith. Hence the construction of the regular solids does not appear as the crowning of a strictly Platonic endeavor, but rather as the result of an Aristqtelian effort to rationalize the very concepts obtained after a first abstraction from the solids or bodies presented to us by the external world. The intimate connection between this effort and the use of classes in the structural developement of the Elements is a further indication of a prevailing Aristotelian intention of influence. In conclusion, it can be truly said that the work of Euclid is situated in the Aristotelian rather than the Platonic perspective, although it 'incorporates the methodological results of both. The rule of logic, already strengthened by the Platonic dialectic of numbers became supreme with Euclid who submitted geometry to its canons. As a consequence, the place given· by Aristotle to geometry in the classification of the sciences strictly adhered to until Descartes and even to the birth of modern analysis. Indeed, the intimate combination of logic and geometry made of the latter almost a real science: by EUCLIJ[) AND ARISTOTLE 403 showing or constructing its objects Euclidian geometry was considered as the real picture of all motion in the universe, up to the present century, though Euclid does not venture to propose his mathematical system as the actual stuff of reality. Hence, it is true to say that the Alexandrian mathematician stamped with an Aristotelian seal the whole system of Greek mathematics. For his work is a factual illustration of a qualitative interpretation of quantity, rather than of a quantitative interpretation of quality, which is a Platonic ideal inherited from the Pythagoreans, and a debatable ambition of modern philosophy. THOMAS University of Montreal, 11iontreal, Canada. GREENWOOD CHRISTIAN LIBERTY [Conclusion] II. CAUSES OF CHRISTIAN LIBERTY i\. MONG the causes to be discussed, the first three, i. e., fl. the Holy Trinity, Jesus Christ the God-Man, and the Catholic Church, are in the strictest sense causes external to the freedom of the Christian soul and produce Christian liberty as efficient causes. Though they are inseparable from true liberty of soul, they are not formal elements integrated in the structure of Christian freedom. But faith, charity, and all the other virtues, infused and acquired, which are treated here as agents in the liberation of the Christian, are, from another viewpoint, essential elements formally constituting Christian liberty. 103 In other words, they have the character of formal cause as well as efficient causality in relation to Christian freedom. The acquired and infused virtues, as human or divine perfections of the potencies they determine, are forms, accidental forms; as operative habits, however, ordered to acts ,_ 04 whereby Christian freedom may be developed, they are agents in the increase of freedom. It is in this latter sense that we speak of them here; we have previously considered the virtues in so far as they are formal constituents of Christian liberty/ 05 Cf., Summa Theol., II-II, q. a. 8, ad !'l-3. Ibid., I-II, q. 49, a. 3. 105 Habitual grace is essentially formal in its causality in relation to Christian freedom. "Grace, as a quality, is said to act upon the soul not after the manner of an efficient cause, but after the manner of a formal cause, as whiteness makes a thing white." (I-ll, q. 110, a. !'l, ad 1.) It is a form divinizing the essence of the human soul. In so far as it operates, mediately, through the supernatural habits which flow from it into the powers of the soul, habitual grace can be considered as active also in the order of efficient causality. (Cf. I-II, q. 110, a. 4, ad 1; ad In treating of the Christian virtues, therefore, we are treating by implication of the efficiency of grace, which is the principle of supernatural action through the medium of the virtues. 108 104 404 CHRISTIAN LIBERTY 405 Law is an efficient cause of freedom in its Christian fulness because it is an imperium, an impulse to action. The New Law, however, in the sense that it is identified with charity, is a formal element (or rather, the formal element) in Christian freedom; it has then whatever is to be attributed to charity in the order of efficient causality. Briefly, the following section deals with the chief causes of Christian liberty not only in its genesis but also in its development and perfection. From this viewpoint it is possible to consider the formal elements in Christian freedom as also operative, effectively, in the increase of that freedom. l. The Holy Trinity " ·where the Spirit of the Lord is, there is freedom," 106 is the formula of St. Paul for the liberty of the Christian. 107 The Holy Trinity is the first cause of all liberty of free creatures whether that liberty be potential, actual, or habitual, whether it be the natural freedom of the will, or the supernatural freedom of grace and glory, whether it be purely internal or manifested externally. 108 But the supernatural liberation of man, though an operation ad extm of the divine nature and therefore a work of the three persons, is appropriated to the Holy Spirit, since it is essentially a divine labor of love and sanctification. 109 It is also a supereminent expression of the divine liberty of the Holy Ghost, " who divides to everyone according as He will." 110 When St. Paql writes elsewhere, " the charity of God is poured forth in our hearts by the Holy Spirit who has been II Cor. 3: 17. Though St. Thomas admits in his commentary on this text that "Lord " may be interpreted as referring directly to Christ, and " spirit " may be understood as relating to the New Law, the spiritual law of Christ, he also gives the interpretation which refers "Spirit" to the Third Person. In II ad Cm·., ca_p. 3, lect. 3: In other places St. Thomas consistently interprets this text as referring to the Holy Ghost as, e. g., IV Cont. Gent., cap. i