Rather than solve problems or simply quantify variables, we consider experiments to be activities that are planned and executed so as to lead to understanding of observed phenomena [2]. Understanding typically means establishing relations, usually in quantitative forms, among observed variables. In particular, an experiment involves these five steps:

1. Identifying a theoretical context that motivates and provides structure for the experiment. The theoretical context establishes possible relations among certain quantities under certain conditions, and it leads to certain questions that can only be resolved by the experiment.
2. Designing and constructing a physical situation—usually an apparatus—that is consistent with the theoretical context and that should be capable of addressing the question.
3. Observing, which usually means measuring, although qualitative observations may also be used. Data are collected.
4. Interpretating the data, which involves assigning meaning to the data, drawing inferences, and making judgments. Meaning provides physical explanations for the relations that are found among quantities. The interpretation may validate the original theoretical context or reveal the limitations of that context.
5. Communicating results.

Thus, measurement and experimental problem-solving are often aspects of an experiment, but an experiment also requires a theoretical context for motivation and interpretation.

2. The Ancients (before 1000 AD)

The development of experimental science is traditionally attributed to Galileo (1564–1642) and refined by Newton (1642–1727). In this section we briefly review and interpret accomplishments of those before Galileo so as to help us address this question: What attitudes changed in the 17th century that promoted the development of an experimental science?

It is sometimes maintained that the ancient Babylonians, Egyptians, Greeks, et al. had no science, that they were primitive peoples, confused and superstitious about the ways in which the world works. In light of the several amazing artifacts that have come down to us, this attitude seems to be a drastic oversimplification. The accomplishments of which we have some knowledge extend over the range of written history, that is to say, over about 5,000 years. (Neugebauer postulates that writing probably originated in Mesopotamia and Egypt around 3,200 BC [3].) A time-table of selected events is presented in Table 1.

Table 1. Time Line for Selected Events Before Newton

Compiled from Neugebauer [3] and Platt [4]

But although the ancients accomplished and mastered certain kinds of knowledge, they failed to develop an experimental component to their studies. What they did accomplish seems to be divisible into three kinds of activities:

1. Ancient engineering, or more properly, experimental problem solving that produced such impressive feats as the Egyptian pyramids (c.2500 BC), the war machines of Archimedes (287–212 BC), Roman bridges, roads, and aqueducts (c.1st century BC), and the machines, toys, and automata of Heron at Alexandria (1st century AD). These accomplishments were largely motivated by political and religious concerns; they did not lead to a strategy for systematic learning about the world.
2. Pure empiricism, that is, systematic measurements that produced quantities of data sufficiently vast that they could be correlated—reduced to models. The models were then used for predictions. The best examples are the astronomical observations of the Babylonians and Ptolemy. The Babylonians used purely arithmetic means for interpreting their data to compose ephemerides for the moon and five known planets; thus, they were able to predict the times of first and last sightings of those bodies, as well as lunar eclipses [3]. In contrast, Ptolemy devised elaborate geometric models of planetary motions in terms of cycles and epicycles by which he also constructed lunar and planetary ephemerides [3]. We view these accurate empirical models as unscientific—not because they were wrong—but because they were not formulated nor tested within a theoretical context.
3. Speculative science, in which the controlling idea is that we can learn about the world only by pure thought and reasoning—experiment and calculation are deceptions to be avoided. These attitudes are exemplified by the writings of Plato and Aristotle and influenced thinkers down into the Middle Ages [6]. Over the centuries, all manner of speculations were propounded, so we should not be surprised that some few of them turned out to be right. For example, in the 5th century BC Democritus (following others before him) speculated about an atomic constitution of matter.

Why did these speculators disdain measurement and calculation? For several reasons, but here are two important ones [6]: First, some things obviously come to us in discrete entities, so we can naturally count them. Thus, we have fingers on a hand, cattle in a herd, and pebbles on the ground. The Latin word calculus meant pebbles for counting. But many other things are continuous, not discrete—things like time, the heat of day, the speed of a runner, and the hardness of materials. The ancient mind did not see how such things could be made discrete, hence countable. Moreover, these continuous quantities were often classified with things that we still do not recognize as measurable—things like beauty, goodness, and justice.

To measure continuous quantities, we must (a) divide them into discrete chunks and (b) identify an arbitrary unit chunk as a standard for the measurement. The ancient attitude tended to be that such arbitrary divisions were unnatural (true) and therefore they could not lead us to deeper understanding of the natural world (false) [6].

A second stumbling block seems to have been the ancient separation of mensuration from pure mathematics. Mensuration is the counting and measurement used in commerce, surveying, the military, and taxation. It led to the geometry of Euclid (c.300 BC), which is the great example of deductive reasoning in action.

Pure mathematics (number theory is a modern example) involves the manipulation of abstract symbols for no practical purposes. The ancients recognized pure numbers as abstract symbols, but they confused the levels of abstraction, and so attributed religious or mystical meanings to numbers. An extreme example of numerical mysticism is provided by the followers of Pythagoras (580–500 BC), who asserted that all is number, that objects are all form and no substance, that purely abstract concepts correspond to particular numbers (e.g., justice was identified with the numeral 4). Plato and Aristotle were much less extreme, but they still doubted that a one-to-one correspondence could be established between numbers and physical objects.

In the ancient pure mathematics, numbers were not used to count or enumerate, but rather were intended to stand as certain mystical symbols and to create certain kinds of impressions. This ancient attitude was not merely one of poetic effect; rather, it betrays a different—indeed alien—mind-set regarding the symbolic value and meaning of numbers [6]. Thus, you must be careful not to project our modern concepts of number onto the numerals appearing in ancient texts. Residuals of these attitudes still appear today, such as in the properties associated with the numeral 13, the numeral 666, and the 40 days of the Old and New Testaments.

In many ancient societies, the distinction between mensuration and pure mathematics was often blurred to serve personal or political ends. The knowledgeable people of ancient societies often tried to protect their advantages by appealing to mystical or magical powers. For example, in ancient Egypt the ability to count on the fingers was considered to be magical [3], and hence the procedure was kept secret and revealed only to the properly initiated.

3. The Middle Ages (1000–1600 AD)

In the Middle Ages ideas concerning the quantification of continuous variables began to change. As usual, the changes were slow, nonuniform, sporadic, and often implicit, though they were driven by definite concrete needs. For example, as late as the 15th century, cooking recipes in England rarely contained definite quantities nor any numbers. An example appears in Figure 1.

Samon roste in Sauce

• Take a Salmond, and cut him rounde, chyne and all,
•      and roste the pieces on a gredire;
• And take wyne, and powder of Canell, and draw it through a streynour;
• And take smale myced oynons, and caste there-to, and let hem boyle;
• And then take vynegre, and pouder ginger, and cast there-to;
•      and then ley the samon in a dissh, and cast the sirip theron al hote,
•      and serue it forth.

Figure 1. Typical English cooking recipe from the 15th century.

Note the absence of definite quantities and numbers. From Black [7].

To illustrate how attitudes toward a continuous variable evolved into a discrete quantification, let's consider the case of time. To the ancients, time embodied two important characteristics [6]: (a) It was continuous, so early time keepers utilized flows: water in the ancient clocks of the Chinese and Archimedes and sand in the hour glasses of the Middle Ages. (b) It was cyclic; thus, day follows day, month follows month, season follows season, and the stars rotate about a pole. Time as a repeating cycle was a psychologically comforting interpretation to the ancients. The interpretation of time as an unfolding of a linear one-dimensional quantity is decidedly modern.

The shift from time continuous to time discrete can be dated to Benedictine monks of the 10th and 11th centuries. First, the monks had the obligation to observe the canonical hours of each day. In different centuries and different countries and different seasons, the number and times of the canonical hours varied, but a typical example would be this:

Without clocks, how did the monks identify 3:00 am for Matins? By the simple expedient of counting: one monk stayed up and counted from bedtime to 3:00 am; a second monk stayed up to keep the counter awake. In this way was time made explicitly discrete.

The Benedictines also made a second contribution to the discretization of time; this was driven by their invention of the Gregorian chant and the need to remember the chants and teach them to novices. Thus, in the 11th century a Benedictine choirmaster invented the first x-y plot [6]:

On this plot, the symbols (notes) used for pitch had finite—discrete—durations. Thus, time was made discrete on a written record, 100 years before the invention of a mechanical clock [6].

Furthermore, the notation for pitch including symbols for discrete rests: a finite duration during which no sound is made. This important invention—a symbol for nothing but holding a place—is fully equivalent to the invention of the zero in the Hindu-Arabic number system.

Once it was realized that the continuous could be made discrete, the idea was taken to extremes. Thus, the schoolmen (literally, scholars at Oxford and Paris) set up scales for measuring goodness, justice, and the like. A modern version of this logical trap is to apply the calculus of cardinals to ordinals. The cardinals are the numbers of counting (1, 2, 3, . . .), while the ordinals establish rank or order (1st, 2nd , 3rd, . . .). Most of calculational procedures routinely used on cardinals have no meaning when applied to ordinals. For example, there is no meaning to the average finishing positions of the horses in a race. Likewise, there is no meaning to the average IQ of any group of people [8].

4. The 17th Century

And so we come to the 17th century with these important, but unconnected, mental attitudes:

1. A speculative science for creating hypotheses (Plato, Aristotle)
2. An empiricism that attributes value to measurement and models (Ptolemy)
3. Deductive reasoning for learning new things from accepted things (Euclid)
4. Experimental problem-solving for applying knowledge (Archimedes, Heron)
5. Quantification for making continuous variables countable (Benedictine monks)

By the 17th century these attitudes were all in place, it merely remained for someone to put them together in a productive way. This, it seems, was the contribution of Galileo and Newton. The combination of these five activities forged a new approach to learning about the world: experimental science.

Literature Cited

[ 1 ] P. Caws, "Definition and Measurement in Physics," in Measurement: Definitions and Theories, C. W. Churchman and P. Ratoosh, eds., Wiley, New York, 1959.

[ 2 ] J. B. Conant, "Foreword," The Early Development of the Concepts of Temperature and Heat, D. Roller, ed., Harvard Case Studies in Experimental Science, Case 3, Harvard University Press, Cambridge, MA, 1950.

[ 3 ] O. Neugebauer, The Exact Sciences in Antiquity, Harper & Brothers, New York, 1962.

[ 4 ] R. Platt, Smithsonian Visual Timeline of Inventions, Dorling Kindersley, London, 1994.

[ 5 ] P. James and N. Thorpe, Ancient Inventions, Ballantine Books, New York, 1994.

[ 6 ] A. W. Crosby, The Measure of Reality, Cambridge University Press, New York, 1997.

[ 7 ] T. Austin, ed., Two Fifteenth-Century Cookery Books, Oxford University Press, 1964; reprinted in M. Black, The Medieval Cookbook, Thames and Hudson, New York, 1992, p. 104.

[ 8 ] J. Ziman, Reliable Knowledge, Cambridge University Press, Cambridge, 1978.