If two chords of a circle intersect anywhere. at any angle, what can be said about the segments of each cut off by the other? The data seem to be too for any conclusion, yet an important and far reaching theorem can be formulated from only this meager amount of “given”:
If two chords intersect, the product of the segments of the one equals the product of the segments of the other.
This theorem says that in Fig. 3A (attached as a jpeg picture), (where the dot is the symbol for algebraic multiplication). What do we mean when we talk about “the product of two line segments”? (There is a way to multiply two lines with a ruler and a compass, but we won’t discuss it here.) What the theorem means is that the product of the respective lengths of the segments are equal. Whenever we put a line segment like PA into an equation, we shall mean the length of PA. (Although we shall soon have to come to grips with the idea of a “negative length” for the present we make no distinction between the length of PA and the length of AP; they are the same positive number).
The Greek geometers took great care to enumerate the different cases of the above theorem. Today, we prefer, when possible, to treat all variants together in one compact theorem. In Fig 3A, the chords intersect inside the circle, in 3B outside the circle, and in 3C one of the chords has become a tangent. Theorem 2 holds in all three cases, and the proofs are so much alike that one proof virtually goes for all.
You may object, and say that in Fig. 3B the chords don’t intersect. But, they do when extended, and in these blogs, we shall say that one line intersects a second when it in fact only intersects the extension of the second line. This is just part of a terminology, in quite general use today, that may be slightly more free-wheeling than what you have been accustomed to. In the same vein, P divided AB internally in Fig 3A, whereas in Fig 3B, P is said to divide the chord AB externally, and we still talk about the two segments PA and PB.
To prove theorem 2, we need to construct two lines, indicated in Fig 3A. Perhaps, you should draw them also in figures 3B and 3C, and, following the proof letter for letter, see for yourself how few changes are required to complete the proofs of these diagrams. In Fig 3A, because they are inscribed in the same circular arc. And, . Therefore, triangles PCA and PBD are similar, and hence have proportional sides:
Till then, aufwiedersehen,
One theorem, the Division Algorithm, acts as the foundation stone upon which our whole development of Number Theory rests. The result is familiar to most of us; roughly, it asserts that an integer a can be “divided” by a positive integer b in such a way that the remainder is smaller than b. The exact statement is given below:
Given integers a and b, with , there exist unique integers q and r satisfying
The integers q and r, are called, respectively, the quotient and remainder in the division of a by b.
We propose to show that the expression is an integer for all . According to the Division Algorithm, every a is of the form 3q, , or . Assume the first of these cases. Then,
, which clearly is an integer. Similarly, if , then
and is an integer in this instance also. Finally, for , we obtain
an integer once more. Consequently, our result is established in all cases.
Inegral numbers are the fountainhead of all mathematics. —- H. MINKOWSKI.
Before becoming weighed down with detail, we should say a few words about the origin of number theory. The theory of numbers is one of the oldest branches of mathematics; an enthusiast, by stretching a point here and there, could extend its roots back to a surprisingly remote date. Although it seems probable that the Greeks were largely indebted to Babylonians and ancient Egyptians for a core of information about the properties of natural numbers, the first rudiments of an actual theory are generally credited to Pythagoras and his disciples.
Our knowledge of the life of Pythagoras is very scanty, and little can be said with any certainty. According to the best estimates, he was born between 580 and 562BC on the Aegean island of Samos. It seems that he studied not only in Egypt, but may even have extended his journeys as far east as Babylonia. When Pythagoras reappeared after years of wandering, he sought out a favorable place for a school and finally settled upon Croton, a prosperous Greek settlement on the heel of the Italian boot. The school concentrated on four mathemata, or subjects of study: arithmetica (arithmetic in the sense of number theory rather than the art of calculating), harmonia (music), geometria (geometry), and astrologia (astronomy). This four field division of knowledge became known in the Middle Ages as the quadrivium, to which was added the trivium of logic, grammar and rhetoric. These seven liberal arts came to be looked upon as the necessary course of study for an educated person.
Pythagoras divided those who attended his lectures into two groups — the Probationers (or listeners) and the Pythagoreans. After three years in the first class, a listener could be initiated in the second class, to whom were confided the main discoveries of the school. The Pythagoreans were a closely knit brotherhood, holding all worldly goods in common and bound by an oath not to reveal the founder’s secrets. Legend has it that a talkative Pythagorean was drowned in a shipwreck as the gods’ punishment for publicly boasting that he added the dodecahedron to the number of regular solids enumerated by Pythagoras. For a time, the autocratic Pythagoreans succeeded in dominating the local government in Croton, but a popular revolt in 501BC led to the murder of many of its prominent members, and Pythagoras himself was killed shortly thereafter. Although the political influence of the Pythagoreans was thus destroyed, they continued to exist for at least two centuries more as a philosophical and mathematical society. To the end, they remained a secret order, publishing nothing and, with noble self-denial, ascribing all their discoveries to the Master.
The Pythagoreans believed that the key for an explanation of the universe lay in number and form, their general thesis being that “Everything is Number.” By number, they meant of course a positive integer. For a rational understanding of nature, they considered it sufficient to analyze the properties of certain numbers. Pythagoras himself, we are told “seemed to have attached supreme importance to the study of arithmetic, which he advanced and took out of the realm of commercial utility.”
The Pythagorean doctrine is a curious mixture of cosmic philosophy and number mysticism, a sort of supernumerology that assigned to everything material or spiritual a definite integer. Among their writings, we find that 1 represented reason, for reason could produce only one consistent body of truth; 2 stood for man and 3 for woman; 4 was the Pythagorean symbol for justice, being the first number that is the product of equals; 5 was identified with marriage, because it is formed by the union of 2 and 3; and so forth. All the even numbers, after the first one, were capable of separation into other numbers, hence, they were prolific and were considered as feminine and earthy — and somewhat less highly regarded in general. Being a predominantly male society, the Pythagoreans classified the odd numbers, after the first two, as masculine and divine.
Although these speculations about numbers as models of “things” appear frivolous today, it must be borne in mind that the intellectuals of the classical Greek period were largely absorbed in philosophy and that these same men, because they had such intellectual interests, were the very ones who were engaged in laying the foundations for mathematics as a system of thought. To Pythagoras and his followers, mathematics was largely a means to an end, the end being philosophy. Only with the founding of the School of Alexandria do we enter a new phase in which the cultivation of mathematics was pursued for its own sake.
It was at Alexandria, not Athens, that a science of numbers divorced from mystic philosophy first began to develop. For nearly a thousand years, until its destruction by the Arabs in 641AD, Alexandria stood at the cultural and commercial center of the Hellenistic world. (After the fall of Alexandria, most of its scholars migrated to Constantinople. During the next 800 years, while formal learning in the West all but disappeared, this enclave at Constantinople preserved for us the mathematical works of the various Greek schools.) The so-called Alexandrian Museum, a forerunner of the modern university, brought together the leading poets and scholars of the day; adjacent to it there was established an enormous library, reputed to hold over 700,000 vokumes — hand-copied at its height. Of all the distinguished names connected with the Museum, that of Euclid (circa 300BC), founder of the School of Mathematics, is in a special class. Posterity has come to know him as the author of the Elements, the oldest Greek treatise on mathematics to reach us in its entirety. The Elements is a compilation of much of the mathematical knowledge available at that time, organized into thirteen parts or Books, as they are called. The name of Euclid is so often associated with geometry that one tends to three of the Books were devoted to Number Theory.
Euclid’s Elements constitutes one of the great success stories of world literature. Scarcely, any other book save the Bible has been more widely circulated or studied. Over a thousand editions of it have appeared since the first printed version in 1482 and before its printing, manuscript copies dominated much of the teaching of mathematics in Western Europe. Unfortunately, no copy of the work has been found that actually dates from Euclid’s own time, the modern editions are descendants of a television prepared by Theon of Alexandria, a commentator of the 4th century AD.
The owner of a drive-in theatre has been professionally informed on the optimum angle that the screen AB should present to the viewer (Figure 1). But, only one customer can sit in the preferred spot V, directly in front of the screen. The owner is interested in locating other points, U, from which the screen subtends the same angle .
The answer is the circle passing thru the three points A, B and V and the reason is:
An angle inscribed in a circle is measured by half the intercepted arc.
Inasmuch as is measured by the same arc as , the angle at U is the same as the angle at V. (If you want a proof of the theorem, just let me know).
A typical example of a calculus problem is this: Of all triangles on the same base and with the same vertex angle, which has the greatest area? We are able to steal the show briefly from calculus and solve this problem almost at a glance, with the aid of Theorem 1. If AB is the base and the given vertex angle is , then all the triangles have their vertices lying on the circle of Figure 1. But, .
One half the base is a constant, and the altitude (and hence, the area) is greatest for AVB, the isosceles triangle.
Even to set up the problem for the calculus approach is awkward (try it and see!) and several lines of careful calculations are required to solve it.
From Theorem 1, we also have the useful corollary that any angle inscribed in a semicircle is a right angle (measured by half the 180 arc of the semicircle). If by any chance we can show that some angle ACB (Fig 2) is a right angle, then we know that a semicircle on AB as a diameter must pass through C. This is the converse of the corollary.