In the era bounded by Galileo's Dialogo of 1632 and Newton's Principia of 1687, science changed. Observation, even when performed with enough care to be called experimentation, gave way to rigorous mathematical analysis as the primary approach to physical phenomena. Whereas Galileo aimed to instruct laymen about his view of the world order by means of plausible arguments and analogies, only an experienced mathematician could hope to understand the world picture envisioned by the Principia. This mathematization of physics was a defining element of that intellectual upheaval we call the Scientific Revolution, and the requirement, still imposed today, that a theoretical physicist be an able mathematician stems from a tradition that flowered in the seventeenth century.

Such sweeping change cannot be attributed to one particular moment or person. Yet the development of this interrelationship between mathematics and physics has remained too long in the realm of vague generalizations, whose validity has yet to be substantiated by a careful comparison with actual events. A new difficulty arises, however, because the particulars against which any generalization must be tested are not well documented. It is the latter deficiency that this book addresses by focusing on a specific person and event in the development of mathematical physics during the seventeenth century. This is a modest endeavor, designed not to explain the greater phenomenon but to provide a case study that any general account must encompass. The person is Christiaan Huygens; the event is his creation of the theory of evolutes.

Preeminent mathematician, physicist, and astronomer, Christiaan Huygens (1629–95) was one of the major figures of the Scientific Revolution.

Evolution

It is fitting that a book based on an evolutionary model should conclude, as it began, with reference to the work of Charles Darwin. Although Darwin never considered applying his evolutionary theory to technology, a number of Darwin's contemporaries readily drew analogies between the development of living beings and material artifacts. The earliest, and perhaps most famous, nineteenth-century figure to do so was Karl Marx, who published his Capital in 1867, eight years after the appearance of Darwin's Origin of Species. Marx's evolutionary analogy includes two stages. In the first stage technology engages humanity in a direct, active relationship with nature. Men and women use their labor to shape physical reality, thus creating the artifactual realm. Once the natural world is transformed by work, nature becomes a virtual extension of the human body. Thus, men and women working with natural objects and forces bring nature within the sphere of human life.

Having minimized the differences between the made and the living worlds, Marx moves on to the second stage of his argument and suggests that the Darwinian approach to the “history of Nature's Technology” be transferred to the “history of the productive organs of man.” He argues that evolutionary explanations should be applied to the organs that plants and animals rely upon for survival and to the technological means that humans use to sustain life.

Because the bob of a freely swinging pendulum traces out a circular arc, which is not isochronous, Huygens had to adjust its swing in such a manner that the pendulum could beat equal times irrespective of the amplitude of its swing. The problem was not new to him. In 1657, when he first applied the simple pendulum to a clock, Huygens was fully aware of its nonisochronous nature for swings of large amplitude; and although at that time he did not know the path that the bob should follow in order to achieve isochronism, he was able to compensate for the irregularity of the simple pendulum by confining its swing between two curved metal bands whose shape he determined empirically. As he explained later to Pierre Petit:

Any attempt to reconstruct the process by which a great mind discovered an important concept becomes itself a process of discovery, and the recounting of the reconstruction becomes a mystery tale twice told. The detritus of the subject's activities – the calculations in the margins, the remains of former values poking out from behind ink blotches – these are the clues left to the historical detective. Shall I tell you of the manuscript that had to exist, although there was no trace of it in the collected works, simply because the derivation it contained was the “logical” next step? Eureka, I found it – or rather he did.

It is incumbent upon me to explain the way that I present the evidence gathered during my pursuit of the elusive Chr. Huygens. For obvious reasons I have adhered as much as possible to his original derivations and avoided the finished propositions. I have rarely presented his arguments verbatim, however, and this raises the possibility that I have misinterpreted what is on the page. In particular, for the sake of rendering the arguments more accessible, I have made some concessions to readability. Modern notation for such entities as pi and the equal sign are introduced, except in one case where I wish to make a point regarding the notation. A potentially more serious substitution is my use of algebraic notation in geometric arguments. Huygens stated everything verbally when he was in his “geometric mode” and used symbols such as the radical sign only when he switched to his “algebraic mode.”

Christiaan Huygens's superior abilities in mathematics and mechanics were already evident in 1646, when he was but seventeen years old. Two examples of his scientific maturity are particularly significant for his future development. The first is a study of the catenary, which Huygens most likely undertook after having read Albert Girard's annotated edition of Simon Stevin's mathematical works, one of the books recommended to Huygens by his teacher as worthy of study. In this work, Girard erroneously claims that the curve formed by a chain hung from both ends, called the catenary, is a parabola. By viewing the hanging chain as a discrete set of equal weights distributed uniformly along a parabola, Huygens was able to disprove Girard's claim.

At about the same time, Huygens undertook the study of a work by Juan Caramuel Lobcowitz, who claimed that the distances traversed by a falling body are proportional to the times elapsed. Huygens refuted Lobcowitz's claim, showing by means of an arithmetic progression that, instead, the spaces are proportional to the squares of the times. In a letter dated September 3, 1646, he boasts to his brother of his discovery, also claiming he can prove that, if the body is projected to one side instead of straight down, it will describe a parabola as it falls: “Of all this and infinitely more things that depend on it I have never known a demonstration before that of the discoverer, myself.” In a postscript the proud discoverer suggests that his brother show the letter to their father.

Diversity

The rich and bewildering diversity of life forms inhabiting the earth has intrigued humankind for centuries. Why should living things appear as paramecia and hummingbirds, as sequoia trees and giraffes? For many centuries the answer to this question was provided by the creationists. They claimed that the diversity of life was a result and expression of God's bountiful nature: In the fullness of his power and love he chose to create the wonderful variety of living things we encounter on our planet.

By the middle of the nineteenth century, and especially after the publication of Charles Darwin's Origin of Species in 1859, the religious explanation of diversity was challenged by a scientific one. According to this new interpretation, both the diversification of life at any given moment and the emergence of novel living forms throughout time were the result of an evolutionary process. In support of Darwin's theories, biologists have proceeded to identify and name more than 1.5 million species of flora and fauna and have accounted for this diversity by means of reproductive variability and natural selection.

Another example of diversity of forms on this earth, however, has been often overlooked or too readily taken for granted — the diversity of things made by human hands. To this category belongs “the vast universe of objects used by humankind to cope with the physical world, to facilitate social intercourse, to delight our fancy, and to create symbols of meaning.”

What, finally, can be said about the man and his method and about their significance? The pattern by which he worked is clearly evident. He was usually spurred to action by a problem posed by another, particularly if solving that problem carried prestige. Once undertaken, it was usually pursued systematically and relatively rapidly to its solution, at which point the salient results were summarized in a short treatise. Because of his thoroughness, Huygens's results often held much broader significance than the original problems from which they stemmed, but they were seldom linked together into a unified theory or applied to new areas, unless resparked by other researchers.

Examples of motivation from the outside abound in Huygens's work. Although his stature with Mersenne had been determined in part by his proof that the catenary was not a parabola, he never returned to the hanging chain to find its true shape until challenged by Jakob Bernoulli. Likewise, he never pursued an inconsistency that he had noted in Bernoulli's analysis of inflection points until he was questioned by 1'Hospital. In still another case, having come to a different conclusion than Bernoulli regarding the sail curve, he asked l'Hospital if he believed Bernoulli's results, concluding, “Your authority would set me to repeat the examination.” Otherwise, it must be presumed, he would not bother to redo his calculations, since there was no guarantee that the effort would “pay.”

Obviously, someone who depended so heavily upon outside stimuli could hardly adhere to a conscious research program.

In this chapter, examples from economics, anthropology, and history illustrate a wholly different set of explanations for the emergence of technological novelty. Of these new explanations, the ones based on socioeconomic factors are the best known and the most fully developed. Their popularity and advanced state of development stem from their connection with economic theory and the Marxist interpretation of historical change. Despite the sophisticated theories and empirical findings that can be marshaled in support of socioeconomic explanations, their drawbacks become evident under critical scrutiny. Therefore, in the end we turn to a broad-based interpretation of innovation that stresses cultural attitudes and values.

Making Things by Hand

“All imitation,” writes social anthropologist H. G. Barnett, “must entail some discrepancy.” No matter how dedicated a copyist is faithfully duplicating an original, the copy always differs from its model. This is true even when the copyist and the original maker are one and the same person; the mindset, materials, tools, and working conditions are all slightly different and that makes exact reproduction impossible. When more people are involved in the copying process, the number of deviations from the original are even greater.

The impossibility of imitation without discrepancy also holds true for mass-produced artifacts. Random variations are admittedly quite small, but they do exist despite the rigid controls employed by modern industry.

The process by which a novel artifact is selected for replication and inclusion into the life of a people involves various factors, some more influential than others. The previous chapter concentrated on economic and military necessity as selecting agents but in the case of the waterwheel, nuclear reactor, and supersonic transport other forces, notably social and cultural ones, were clearly at work. Ancient and medieval religious beliefs, a bias toward the acceptance of advanced technology, and Utopian energy myths each played an ancillary role in the selection of these innovations. In this chapter the social and cultural factors governing selection will be raised to a position of central concern and examined with the aid of cross-cultural comparisons.

Technology and Chinese Culture

The influence of cultural values and attitudes on technological choices is more readily apparent in examples drawn from remote cultures than from those that share our Western outlook. Technology is so intimately identified with the cultural life of a people that it is difficult for an indigenous observer to gain the objectivity necessary for critical appraisal. Fortunately, Chinese history contains a wealth of material on technology and culture, much of which has been studied by modern Western historians. Therefore, it is of Chinese civilization that we first ask the question, How may culture affect the selection and replication of technological novelties?

Often responding to questions stated by others and exceedingly proud of his ability to find the answers, Huygens obviously desired the acclaim of his contemporaries. He constantly compared himself to the illustrious forefathers of science, undoubtedly searching for the recognition that was theirs. Yet the erratic way in which he communicated his results guaranteed a diffuse and diverse reception and ultimately undermined his place in history.

Certainly, the contemporary response to Huygens's theory of evolutes and its companion discoveries cannot be judged by the notice given to the Horologium Oscillatorium, which was not published until fourteen years after Huygens had created the mathematical technique, by which time most researchers had already learned of its major applications through more informal channels of communication. Indeed, many of the people with whom Huygens had scientific ties heard about the cycloidal clock and the isochronous path of its bob within a year of its creation.

Of course, van Schooten had been told about the new design almost at the moment of its birth, and a month later a similar description went to the Flemish mathematician Andreas Tacquet. However, although both letters contain a sketch of a pendulum hung between curved plates, neither identifies the curve as a cycloid. Instead, the derivation was left as an unspoken challenge to the recipient.

Subsequent letters to Jean Chapelain, Pierre de Carcavy, Boulliau, and Wallis are even more vague, with no sketch and only a brief claim to have created an isochronous clock.

In solving his physical question, Huygens had developed a sophisticated mathematical tool that rightly takes its place in the Horologium Oscillatorium as one of his great discoveries. But just as his physics went beyond the narrow problem of the constant of gravitational acceleration to a broader discussion of accelerated motion, so his mathematics moved beyond the shape of curved plates to an analysis of one of the fundamental questions of seventeenth-century geometry. What, indeed, was the significance of the theory of evolutes as mathematics?

The evolute as a purely mathematical concept has long been relegated to a role in the history of curvature, a role determined by the very definition of curvature. How, indeed, is the bending of a given curve to be measured? It seems obvious that a circle bends uniformly (that is, its curvature is constant) and that a small circle bends more sharply than a large one (that is, its curvature is greater) and, consequently, that the “curvature” of a circle might best be defined as the reciprocal of its radius. For any other curve the bending varies from point to point. Given a specific point, however, the most evident way to measure the bending is to assign to the curve the curvature of the circle that best approximates the curve in the immediate area of the specified point. The best approximating circle (labeled the “osculating circle” by Leibniz) can be derived by drawing normals to the curve at the given point and at a point infinitesimally near to it.