The Concepts of the Calculus by Carl B. BoyerReview by: I Bernard CohenIsis, Vol. 32, No. 1 (Jul., 1940), pp. 205-210Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/226090 .

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REVIEWS 205

have resulted in their rejection by physicists. Professor DINGLER does not give the detailed mechanical explanations of electricitv and light which would be required to win assent for his system. Classical physics is embodied in our measuring instruments, not because it alone is un- equivocal, but because measuring instruments are macrophysical objects from which classical principles were abstracted. The propositions of Euclidean geometry describe the observable positional relations of practically rigid bodies of which measuring instruments are built. These facts have been taken into account in the theories of relativity and quantum mechanics which include the classical theories as limiting cases. BOHR, who has contributed so much to the development and interpretation of quantum theory, has emphasized the fact that all ex- perimental results are interpreted in terms of classical ideas.

Although the reviewer must emphatically reject Professor DINGLER's contention that physics can be built only on classical foundations, he acknowledges the stimulating character of the thesis and the great value of the general analysis. Like BRIDGMAN, who has set forth the operational theory of physical concepts, Professor DINGLER stresses the manual aspect of physics and works out a systematic logic of purpose. The logic of determinations and the logic of hypotheses are also notable contributions. There is an admirable discussion of probability and induction. The function of the criterion of mass in the development of chemistry is clearly explained. In the brief discussion of biology the author relaxes the strictness of his point of view with the remark that whethier or not specific properties are to be found in a given body depends on experience. Throughout the work it is pointed out that illusory metaphysical problems arise from lack of understanding of methodology. It is a pity that Professor DINGLER'S rejection of the fundamental significance of the quantum theory and theory of relativity results in lack of widespread recognition of his learning, constructive imagination, and productivity.

University of Caltformia, Berkeley. V. F. LENZEN.

Carl B. Boyer.-The Concepts of the Calculus. A critical and historical discussion of the derivative and the integral. New York, Columbia University Press, 1939. VI+346 p. $3.75.

The author states (p. v) that " this is not a history of the calculus in all its aspects, but a suggestive outline of the development of the main concepts." As such, it is reallv a history of mathematics from the focal point of the derivative and the integral and Mr. BoYER traces the main ideas of the subject from antiquity to modern times. The book

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2o6 isIs, XXXII, I

represents a prodigious amount of labour, and the richness of material will make it a source book for all mathematicians. Perhaps there are too many footnotes. For example, on p. i it is stated that " no general agreement has been reached as to the nature of the subject [mathematics], nor has anv universally accepted definition been given for it." The- authoritv in the footnote to this statement is BELL, The Queen of the Sciences. It certainly is not necessarv to refer to a popular book for such a universally acceptable and innocuous statement. This example could easily be multiplied many times over. But, to give a statement of LMBNIz a new orientation, such errors of commission are much to be preferred to major errors of omission. The work is divided as follows-Introduction, I3 P., Conceptions in

Antiquity, 47 p., Mledieval Contributions, 35 p., A Century of Anticipation (C.I550 - c.i650), 9I p., NEWTON and LEIBNIZ, 37 p., The Period of Indecision (The eighteenth century), 43 p., The Rigorous Formulation (The nineteenth centurv), 32 p., Conclusion, I2 P. If it seems disproportionate to give the " century of anticipation" more than twice as much space as any of the other sections, it is surely to be remembered that in many ways this was the crucial century for the history of the calculus. For it was during this period that mathematics was developed to such a final form that NEWTON and LEIBNIZ were able to present their algorithmic devices immediately afterward. That is, during this period, we witness the growth and the ;' modem " formulation of analytic geometry (DESCARTES, FERmAT), of algebra as we now conceive the subject (WALLIS, OUGHTRED, HARRIOT), of number theory (STEVIN), and the work of the immediate precursors of the calculus (ROBERVAL, CAVALER, BAImow). Yet the extreme lenght of this chapter must be criticized on stylistic grounds ; this length makes for difficult reading.

By far the most interesting chapter is that on the " medieval contributions," especially in the fine account of the mathematical work of the " Calculator," RICHARD SUISETH, who lived in the first half of the fourteenth centurv. SUISETH is not mentioned in CAJORI's History of Ml'athematics, and SMITH, in vol. I of his History of Ml4athematics, says of him only that " he was educated at Merton College, Oxford, and wrote an obscure work on mathematics. It treats of a subject just beginning to attract attention in England and in France, De latitudinibus formarum." Due to this work he gained the title of " calculator acutissimus." SUISETH (his name has many variations; the insignificant D.N.B. account uses SWINESHEAD) was a Cistercian monk in Swineshead. in Lincolnshire, England, and composed about twelve treatises of which few survive. His best known work is that called the Liber calculationum. Our author establishes that this was written later than 1328, inasmuch

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REVIEWS 207

as it refers to BRADWARDINE'S Liber de proportionibus of that year " and probably dates from the second quarter of the fourteenth century." Mr. BoYER's account is extremely valuable for, as he says, " No analysis of the Liber calculationum from the mathematical point of view appears to be available" although a general description of the contents of the book may be found in DUHEM, Etudes sur LEONARD DE VINCI, VOl. III, P. 477-8I, and THORNDIKE, History of Magic, etc., vol. III, chap. 23. Mr. BoYER points out significantly that DUHEM "w would see in the Liber calculationum the influence of ORESME and would characterize the work of SUISErH as" l'ceuvre d'une science senile et qui commence a radoter," although more recently THORNDIKE has shown the work of ORESME to be subsequent to that of the Calculator. It should be mentioned as well that the idea, due to DUHEM, that Calculator referred in his work. only to BRADWARDINE, ARISTOTLE, and AvERRoES is unfounded. Mr. BoyE has found in the original text the names of many ancient mathematicians, for example EUCLID and BoEmIus.

Mr. BOYER quotes at some long length from the original work (transcribing the abbreviations) since the original is difficult to obtain. Calculator seems to have been interested, above all, in problems of uniform and non-uniform variation. In considering problems of thermal content, for example, he

reached the conclusion... that the average intensity of a form whose rate of change over an interval is constant, or of a form which is such that it is uniform throughout each half of the interval, is the mean of its first and last intensities. The rigorous proof of this requires the use of the limit concept, but Calculator had resort to dialectical reasoning, based on physical experience of rate of change... The methods... are amplified and applied to numerical examples and... to questions dealing with density, velocity, and the intensity of illumination.

One of the most interesting results obtained by him arose from the following problem: if throughout half of a given time interval a variation continues at a certain intensity, throughout the next quarter of the interval at double this intensity, thoughout the following eighth at triple this, and so ad infinitum; then the average intensity for the whole interval will be the intensity of the variation during the second subinterval (or double the initial intensity).

This is clearlv equivalent to the summation of the infinite series

I 2 3 4 n _+ - + +.. + -+...2

2 4 8 i6 2n

As Mr. BOYER remarks, Not only is the time interval in his problem infinitely divided, but the intensity

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2o8 Isis, xxxii, I

itself becomes infinite. Now how can a quantity, whose rate of change becomes infinite, have a finite average rate of change? SuIsErH admitted that this paradoxical result was in need of demonstration and so furnished at great length the equivalent of a proof of the convergence of the infinite series.

I will not bother to sketch out that proof; see the work under review, pp. 77-78.

To sum up with Mr. BOYER:

SuISE}I's unfortunate adoption of the Peripatetic attitude that such qualities as dryness, coldness, and rarity are the opposites of moistness, warnth, densitv-rather than simply degrees of the latter-complicated his consideration of these... Nevertheless, to Calculator we owe perhaps the first serious effort to make quantitatively understandable these concepts of mathematical physics. His bold study of the change of such quantities anticipated not only the scientific elaboration of these, but also adumbrated the introduction into mathematics of the notions of variable quantity and derivative. In fact, the very words fluxus and fluens which Calculator used in this connection, were to be employed by NEWTON some three hundred years later, when in his calculus he spoke of such a variable mathematical quantity as a fluent and called its rate of change a fluxion.

The treatment of OmmSME, CUSA, and other medievals who contributed to the development of ideas in this vein is equally thorough.

It is much to be regretted that the other chapters are not written as carefullv and as clearly as this one on the medievals. For the most part, however, Mr. BoYER keeps distinct early ideas relating to the integral as opposed to those relating to the derivative, although he shows how these are connected. Is it that in the chapter on " medieval contributions", Mr. BOYER has made so much original contribution of his own, that has caused that chapter to be, far and above, the best in the book ?

Unfortunately, at times Mr. BOYER makes very loose statements such as that PASCAL "' was not so much a creative genius as a mathematician, scientist, and philosopher, with a remarkable flair for clarifving ideas which had been somewhat vaguelv set forth by others, and for supplving these with a reasonable basis" (p. I47). This in face of his highly original treatment of conics, and the latter part of the statement in face of some of the religious and mystical ravings concerning mathematics ! Other types of too simple generalization to which Mr. BoYER is too easy prev may be exampled bv the following:

The derivative and integral had their sources in two of the most obvious aspects of nature-multiplicity and variability (p. 3).

The calculus had its origin in the logical difficulties encountered by the ancient Greek mathematicians in their attempt to express their intuitive ideas on the ratios or proportionalities of lines, which they vaguelv regarded as continuous, in terms of numbers, which they regarded as discrete (p. 4).

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REVIEWS 209

Or, examine the statement on p. 209 that " with reference to the logical and philosophical justifications of his procedure... LEIBNIZ was less emphatic than NEwrON... He did not wish to make of the infinitely small a mystery, as had PASCAL; nor did he turn to geometrical intuition for clarification." Now of course LEIBNIZ used geometrical intuition. Even Mr. BoYER will admit that, and in fact, does. On p. 2II, he says: "Perhaps realizing that this definition could not be consistently applied, he later gave a geometrical interpretation..." On p. 2I2, he says " In the absence of satisfactory definitions, LEIBNIZ resorted frequently to analogies to clarify the nature of his infinitely small differentials..." "Appealing again to geometrical intuition, LEIBNIZ said that as a point added nothing to a line,... so differentials of a higher order... may be neglected."

On p. 14 Mr. BoYE remarks that " the pre-Hellenic peoples are usually regarded as prescientific in their attitudes toward nature," adding that " on this point there are significant differences of opinion." These " significant differences of opinion " are represented for the author by BARRY (The scientific habit of thought) and BuRiur (Greek Philosophy, Part I, THALEs to PLATO) as opposed to KARPINSKI (Is there progress in math. discovery? Isis, vol. XXVII, 46-52). According to BuRNzr (with BARRY in accord), " natural science is the creation of the Greeks " and " there is not the trace of that science in Egypt or even in Babylon " while, according to KARPINSKI, the " achievements of the Babylonians, Egyptians, and Hindus are scientific in the highest sense." Is not the reason for this apparent contradiction to be found in the simple fact that BuRNir's work dates from 1914, BARRY'S from 1926, and KARPINSKI'S

from 1937? The fundamental investigations of THUREAU-DANGIN and NEUGEBAUER, on which we base our present knowledge of the scientific work of the pre-Greeks was not accomplished till after the first two works had been publishei !

Thus, HEATH, writing in I931 (A manual of Greek mathematics) says that chief among the differences between this work and his earlier History of Greek mathematics (I92I) is the fact that he has " taken account of the striking and even mystifying results of the study and interpretation of certain ancient Babylonian mathematical texts which have appeared in the last few years, especially in I929 and I930 ". Mr. BOYER'S

bibliography does not include the LVanual, although it does include the History. Further examination of the bibliography is revealing. For the author includes such worthless books as BELL'S The search for truth, and BALL'S History of mathematics (I888) although he does not include BELL'S M11en of mathematics, which contains some very interesting remarks on certain important mathematicians discussed bv Mr. BOYER, nor does

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2IO ISIS, XXII, I

he include BALL'S very useful treatise, A history of the study of mathematics- at Cambridge. It is difficult, too, to explain the presence of HOGBEN'S Sciencefor the citizen; there might be some justification for his Mathematics for the million which is not on the list.

Curiously too, Mr. BOYER used the Opera mathematica of JOHN WALLIS of I656-57 (2 vols.) rather than the later Opera of I693-99 (3 vols.). This has forced Mr. BOYER to make the statement that ' NEWTON, in fact, acknowledged that he had been led to his first discoveries in analysis and fluxions by the Arithmetica infinitorum of WALLIS, and the principles of induction and interpolation which WALLIS there employed may have been instrumental also in leading NEWTON to the discovery of the binomial theorem ". The last part of this statement is attributed by Mr. BOYER to a secondary source. Had he used the later edition, he would have found a letter (vol. III, p.634) from NEWTON to OLDENBURG, in which he states that the idea for the binomial theorem came directly from a study of WALLIS'S work on series.

But for the few occasional misstatements of the kind instanced above, the book is an admirable one, covering a much needed field of investigation. It is to be hoped that in the near future, Mr. BoyER will revise the book to bring it all up to a high standard of excellence. For Mr. BoYER's contributions are highly original and very valuable. If I have attempted to show the pitfalls into which he has fallen, it is only so that the reader may gain the full value of Mr. BoYER's work without being led astray by errors which can easily be rectified.

I BEmARD COHEN.

Gleason L. Archer.-History of radio to 1926. Vr+42I p. New York, American Historical Society, 1938. ($4.00).

The history of radio forms an exciting chapter in the history of thought. When one views the gargantuan industry connected today with radio communication, and when one considers the tremendous sweep of research, both pure and practical, allied with it, it seems hardly possible that " radio broadcasting as we know it today, began as a scientific novelty in East Pittsburgh, Pennsylvania, in the year i920 ."

One of the most interesting features of the book under review is the section devoted to pre-radio communication systems. FRANKLIN devised a means of signalling with electricity and there followed a quick succession of similar schemes, some using magnets, others even making use of litmus solutions. The most notable of these systems were those of JOSEPH BOZOLUS 1767, LOUIS LE SAGE I774, FRANCIS RONALDS I8I6, HARRISON GRAY DYAR 1828, Baron SCHELLING 1823. Although most of these schemes were never even applied for demonstration purposes, it is

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