Archimedes

s and of the significance of his work it is necessary to pass in review the course of develo

ng that geometry was invented by the Egyptians and t

Nor is it marvellous that the discovery of this and the other sciences should have arisen from such an occasion, since everything which moves in the sense of development will advance from the imperfect to the perfect. From sense-perception to reasonin

ypt and thence introduce

rmul?, more or less inaccurate, for certain measurements, e.g. for the areas of certain triangles, parallel-trapezia, and circles. They had, further, in their construction of pyramids, to use the notion of similar right-angled triangles; they even had a name, se-qet, for the ratio of the half of the side of the base to the height, that is, for what we should call the co-tangent of the angle of slope. But not a single ge

f the Great Bear as a means of finding the pole. In geometry the following theorems are attributed to him-and their character shows how the Greeks had to begin at the very beginning of the theory-(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He

cs as a study pursued for its own sake and made it a part of a liberal education. Pythagoras, son of Mnesarchus, was born in Samos about 572 B.C., and died at a great age (75 or 80) at Metapontum. His interests were as various

ntric view of the universe, and it was his successors who evolved the theory that the earth does not remain at the centre but revolves, like the other planets and the sun and moon, about the "central fire". Perhaps his most remarkable discovery was the dependence of the musical intervals on the lengths of vibrating strings, the proportion for the octave being 2 : 1, for the fifth 3 : 2 and for the fourth 4 : 3. In arithmetic he was the first to expound the theory of means and of proportion as applied to commensurable quantities. He laid t

ymnasium, he sought out a well-favoured youth who seemed likely to suit his purpose, and was withal poor, and bribed him to learn geometry by promising him sixpence for every proposition that he mastered. Very soon the youth got fascinated by the subject for its own sake, and Pythagoras

m with the proof of this theorem, we prefer to believe that tradition is right. This is to some extent confirmed by another tradition that Pythagoras discovered a general formula for finding two numbers such that the sum of their squares is a square number. This depends on the theory of the gnomon, which at first had an arithmetical signification corresponding to the geometrical use of it in Euclid, Book II. A figure in the shape of a gnomo

+ ... + (2

at their sum is a square we have only to see that 2n + 1 is a square. Suppose that 2n + 1 = m2; then n = ?(m2 ? 1), and we ha

y of proportionals and the construction of the

ectilineal figure; this construction is the geometrical equivalent of arithmetical division. The general case is that in which the parallelogram, though applied to the straight line, overlaps it or falls short of it in such a way that the part of the parallelogram which extends beyond, or falls short of, the parallelogram of the same angle and breadth on the given straight line itself (exactly) as base is similar to another given parallelogram (Eucl. VI., 28, 29). This is the geometrical equivalent of the most general form of quadratic equation ax ± mx2 = C, so far as it has real roots; while the condition that the roots may be real was also worked out (= Eucl. VI., 27). It is im

ven rectilineal figure and similar to another. Plutarch mentions a doubt as to whether it was this problem

lid, Book II.) were also Pythagorean; the construction of a square equal to a given rectangle (Euc

the sum of the angles of any triangle is

ortion applied only to commensurable magnitudes). Our information about the origin of the propositions of Euclid, Book III., is not so complete; but it is certain that the most important of them were well known to Hippocra

io ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable was bound to cause geometers a great shock, because it showed that the theory of proportion invented by Pythagoras was not of universal application, and therefore that propositions proved by means of it were not really established. Hen

line and circle, and which were destined to play a great part in determining the direction taken by Greek geometry in its highest flights. The main problems in question were three: (1) the doubling of the cube,

ameters (= Eucl. XII., 2), with the deduction that similar segments of circles are to one another as the squares on their bases. These propositions were used by him in his tract on the squaring of lunes, which was intended to lead up to the squaring of the circle. The latter problem is one which must have exercised practical geometers from time immemorial. Anaxagoras for instance (about 500-428 B.C.) is said to have worked at the problem while in prison. The essential portio

ever, the cubical form. According to the other, the Delians, suffering from a pestilence, were told by the oracle to double a certain cubical altar as a means of staying the plague. Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in contin

clusively in the form of the problem of the two mean pro

degree theoretical. It consisted in determining a point in space as the intersection of three surfaces: (a) a cylinder, (b) a cone, (c) an "anchor-ring" with internal radius = 0. (2) Men?chmus, a pupil of Eudoxus, and a contemporary of Plato, found the two mean proportionals by means of conic sections, in two ways, (α) by the intersection of two parabolas, the equations of whi

circle. It was theoretically constructed as the locus of the point of intersection of two straight lines moving at uniform speeds and in the same time, one motion being angular and the other rectilinear. Suppose OA, OB are two radii of a circle at right angles to one another. Tangents to the circle at A and B, meeting at C, form with the two radii t

hear were the following. Leon, a little younger than Eudoxus (408-355 B.C.), was the author of a collection of propositions more numerous and more serviceable than those collected by Hippocrates. Theudius of Magnesia, a contempo

ahedron, cube, octahedron, dodecahedron, and icosahedron), and Euclid was therefore no doubt equally indebted to The?tetus for the contents of his Book XIII. In the matter of Book XII. Eudoxus was the pioneer. These facts are confirmed by the remark o

s predecessors, and it is his achievements, especially the discovery of

pass through the centre of the earth. The outermost sphere represents the daily rotation, the second a motion along the zodiac circle or ecliptic; the poles of the third sphere, about which that sphere revolves, are fixed at two opposite points on the zodiac circle, and are carried round in the motion of the second sphere; and on the surface of the third sphere the poles of the fourth sphere are fixed; the fourth sphere, revolving about the diameter j

expounded in Euclid, Book V., and which still holds its own and will do so for all time. He also solved the problem of the two mean propo

qual magnitudes of the same kind and from the greater you subtract not less than its half, then from the remainder not less than its half, and so on continually, you will at length have remaining a magnitude less than the lesser of the two magnitudes set out, however small it is. Archimedes says that the theorem of Euclid XII., 2, was proved by means of a certain lemma to the effect that, if we have two unequal magnitudes (i.e. lines, surfaces, or solids respectively), the greater exceeds t

rst step. He inscribed a square (or, according to another account, an equilateral triangle) in a circle, then bisected the arcs subtended by the sides, and so inscribed a polygon of double the number of sides; he then repeated the process, and

has the same base and equal height. Archimedes, however, tells us the remarkable fact that these two theorems were first discovered by Democritus (who flourished towards the end of the fifth century B.C.), though he was not able to prove them (which no doubt means, not that he gave no sort of proof, but that he was not able to establish the propositions by the rigorous method of Eudoxus). Archimedes adds that we must give no small share of the credit for these theorems to Democritus; and this is another testimony to the marvellous powers, in mathematics as well as in other subjects, of the great man who, in the words of Aristotle, "seems to have t

s proved by means of the same lemma. The proofs of the propositions about the volumes of pyramids, cone

polygon of eight sides, then similarly we draw the regular inscribed polygon of sixteen sides, and so on. We erect on each regular polygon the prism which has the polygon for base, thereby obtaining successive prisms inscribed in the cylinder, and of the same height with it. Each time we double the number of sides in the base of the prism we take away more than

of prisms inscribed in the cylinder until the parts of the cylinder le

C ? P

? 3V

fore

that P is equal to three times the pyramid wi

the volume of the cone: which is impossi

is not gre

that C < 3V, so

1?

han the excess of V over 1?3 C. It follows that the pyramid is greater than 1?3 C. Hence the prism on the same base as the pyramid and inscribed in the cylind

eater than 1?3 C, or

eater nor less than 3V, must be

rchimedes refers to as the "Elements of Conics". We know of two such treatises, (1) Euclid's four Books on Conics, (2) a work by one Arist?us called "Solid Loci," pro