1564–1642

Galileo Galilei was an Italian astronomer, physicist, and mathematician. “More than any other man, Galileo is the founder of the methodology of modern science.”(1) Prior to Galileo, mathematics was used to solve certain kinds of problems in the physical world, such as measuring the distance between two places or the area of a plot of land. Mathematics had not been used, however, to study objects in motion systematically. Rather than empirical observation and mathematical modeling, the science of physics was based upon the logic of Aristotle. Galileo broke with tradition through his pioneering work in gravitation and motion, which combined mathematical analysis with experimentation. Thus Galileo often is referred to as the founder of modern mechanics and modern physics. Mechanics, in this case, is the branch of applied mathematics dealing with motion and tendencies to motion. “Possibly the most far-reaching of his achievements was his reestablishment of mathematical rationalism against Aristotle’s logico-verbal approach and his insistence that the ‘Book of Nature is ... written in mathematical characters.’ From this base, he was able to found the modern experimental method.”(2)

Mathematics/Physics

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Galileo informally stated the principles later embodied in Isaac Newton’s first law of motion. Newton’s three laws of motion form the foundation of physics. Newton’s first law is:

(1) (Law of Inertia) That a body remains in its state of rest unless it is compelled to change that state by a force impressed on it.

Galileo’s discoveries include the constancy of a pendulum’s swing, later applied to the regulation of clocks. Galileo, through experimentation, discovered the simple law that, “whatever the length of the pendulum swing, the time taken to complete the swing is the same.” Thus the pendulum must accelerate faster the greater the distance it must travel in order to complete a swing in the same amount of time as a shorter swing. Scientists call this the pendulum’s periodic swing. He formulated the law of uniform acceleration of falling bodies. Falling objects (including a pendulum, in this case) gradually gain speed (accelerate) as they fall from a resting position. Previous to Galileo’s discovery, it was assumed objects immediately attained a constant speed after being dropped.

Galileo created a mathematical model for the acceleration of falling objects. Because instruments for measuring time were somewhat crude in the seventeenth century, it was difficult to determine accurately the amount of time that it takes for an object to fall from a given height. Galileo had no watch or clock to measure time for his experiments. The sandglass was one of the only means of measuring time before Galileo’s discoveries. At first, he used his pulse because it was more accurate than any manufactured timepiece of the day; this method was accurate enough to discover the constancy of a pendulum’s swing. For his experiments on falling bodies, he needed greater accuracy, so he set up a large barrel of water next to his experiments. At the base of the barrel there was a small hole that allowed the contents to empty gradually. The water was allowed to drip for the length of the experiment. He repeated the experiment many times and measured the quantity of the water collected in the bucket beneath the barrel. Later this setup was called a “water clock.”(3)

Galileo, a clever mechanical designer, also built polished ramps along which he rolled smooth ball bearings. With the ramps inclined at various angles, he was able to simulate the motion of a falling object. To overcome the lack of an accurate timepiece, he used the “water clock” to accurately measure how far the ball bearing traveled at given times.(4) Amazingly, these were the first timed experiments, and he was the first to use measurement in a systematic way, and the first to apply mathematics to physical phenomena.(5)

Mathematics–Concept of a Function or Relation Between Variables

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From Galileo’s study of motion, mathematics derived a fundamental concept that was central to practically all of the work for the next 200 years—the concept of a function or a relation between variables.(6) One finds this idea throughout Galileo’s Two New Sciences, the book in which he founded modern mechanics/engineering. Two New Sciences is also considered the first popular science book because it was written in Italian, the language of the people, instead of Latin which few people could read. 

Galileo expresses his functional relationships in words and in the language of proportion. He begins with the most fundamental relationship, which is Newton’s second law of motion: The change in motion is proportional to the mass of a body times its acceleration or change in velocity. In his work on the strength of materials, he states, “The areas of two cylinders of equal volumes, neglecting the bases, bear to each other a ratio which is the square root of the ratio of their lengths.” In his work on motion, he states, for example, “The times of descent along inclined planes of the same height, but of different slopes, are to each other as the lengths of these planes.” The language shows clearly that he is dealing with variables and functions. It was but a short step to write these statements in symbolic form. Since the symbolism of algebra was being extended at this time, Galileo’s statement on the distances traveled by a falling body soon was written as d = s t², or in words, distance equals speed times time squared; and his statement on times of descent as t = √ds, or in words, time equals the square root of the distance times the speed.(7)

He also described the parabolic trajectory of projectiles. Before Galileo, people followed Aristotle’s notion that a projectile, such as a cannonball, simply flew in two straight lines and dropped out of the sky above the target. By careful experiments, he showed that a fired cannonball was sent along an arched pathway called a parabola. By repeating his experiments hundreds of times, Galileo showed the greatest range could be obtained if the cannon was angled at 45 degrees to the horizontal.(8)

Astronomy

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Galileo was the first man to use the telescope to study the sky and observed craters on the moon, sunspots, the stars of the Milky Way, Jupiter’s satellites, and the phases of Venus. Most importantly, he amassed evidence that proved the Earth revolves around the sun and is not the center of the universe, as had been believed. His belief in the Copernican system represented a radical departure from accepted thought and was rejected by the Catholic Church, which saw him as a threat to Christian doctrine. He set forth his Copernican worldview in his great book, Dialogo sopra i due massimi sistemi del mondo, tolemaico e copernicano (Dialogue Concerning the Two Chief World Systems—Ptolemaic and Copernican,113 pages, 1632). Under threat of torture from the Inquisition, he publicly recanted his heretical views, but was still forced to spend the last eight years of his life under house arrest.(9)

Mathematics–Logic, Inductive Method

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Galileo affected a revolution by rejecting the Greek method of deduction and introducing the modern method of induction in the sciences. Instead of building conclusions upon an assumed set of generalizations, the inductive method starts with observations and derives generalizations, or axioms. Of course, even the Greeks obtained their axioms from observation; Euclid’s axiom that a straight line is the shortest distance between two points was an intuitive judgment based on experience. But whereas the Greek philosophers minimized the role played by induction, Galileo looked upon induction as the essential process for gaining knowledge, the only way of justifying generalizations. Moreover, he realized that no generalization could be allowed to stand unless it was repeatedly tested by newer and still newer experiments—unless it withstood the continuing test of further induction.

Galileo’s general viewpoint was just the reverse of the Greeks’. Far from considering the real world an imperfect representation of ideal truth, Galileo considered generalizations to be only imperfect representations of the real world. No amount of inductive testing could render a generalization completely and absolutely valid. Even though billions of observations tend to bear out a generalization, a single observation that contradicts or is inconsistent with it must force its modification. And no matter how many times a theory meets its tests successfully, there can be no certainty that it will not be overthrown by the next observation.

Philosophy

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This, then, is a cornerstone of modern natural philosophy. It makes no claim of attaining ultimate truth. In fact, the phrase “ultimate truth” becomes meaningless because there is no way in which enough observations can be made to make truth certain, and therefore “ultimate.” The Greek philosophers recognized no such limitation. Moreover, they saw no difficulty in applying exactly the same method of reasoning to the question “What is justice?” as to the question “What is matter?” The new natural philosophy started by Galileo, on the other hand, made a sharp distinction between the two types of questions. The inductive method cannot make generalizations about what it cannot observe, and since the nature of the human soul, for example, is not observable by any direct means yet known, this subject lies outside the realm of the inductive method.

The new natural philosophy came in time to be called “science” (from the Latin word meaning “to know”). It is generally taken to mean specifically the inductive method. The older term “natural philosophy” lingered on as a synonym for science through the nineteenth century, and it is still recognized in the highest degree given to scientists—the Ph.D., or “Doctor of Philosophy.”(10)

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Footnotes:

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(1) Morris Kline, Mathematical Thought From Ancient to Modern Times (New York, 1972), p. 334.

(2) Encyclopaedia Britannica, Macropaedia, Volume 19, 1993, 15th Edition, p. 638.

(3) Michael White, Galileo Galilei –Inventor, Astronomer, and Rebel (New York, 1999), pp. 3, 18, 19, 35, 38 (from “because instruments”).

(4) Donald R. Franceschetti (editor), Biographical Encyclopedia of Mathematicians (New York, 1999), p. 212.

(5) Isaac Asimov, The Intelligent Man's Guide to Science, Volume 1 - The Physical Sciences (New York, 1960), p. 16.

(6) Kline, p. 338.

(7) Ibid., p. 338.

(8) White, pp. 3, 18, 19, 35, 38.

(9) Encyclopaedia Britannica, Macropaedia, Volume 19, 1993, 15th Edition, pp. 567, 638.

(10) Asimov, pp. 16-18.

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