Sir Thomas Little Heath

“ It is a sufficient answer to this argument that, if Plato really meant in the passage of the Timæus to say that the earth revolves about the axis of the heavenly sphere, he had nothing to repent of. We must therefore, for our part, conclude that in his written Dialogues Plato regarded the earth as at rest in the centre of the universe.
We have it on good authority that Plato set it as a problem to all earnest students “to find what are the uniform and ordered movements by the assumption of which the apparent movements of the planets can be accounted for”. ”

“ But he put forward a remarkable and original hypothesis to explain the Milky Way. He thought the sun to be smaller than the earth. Consequently, when the sun in its revolution passes below the earth, the shadow cast by the earth extends without limit. The trace of this shadow on the heavens is the Milky Way. The stars within this shadow are not interfered with by the light of the sun, and we therefore see them shining; those stars, on the other hand, which are outside the shadow are overpowered by the light of the sun which shines on them even during the night, so that we cannot see them. ”

“ And it is to be feared that few who are not experts in the history of mathematics have any acquaintance with the details of the original discoveries in mathematics of the greatest mathematician of antiquity, perhaps the greatest mathematical genius that the world has ever seen.
History and tradition know Archimedes almost exclusively as the inventor of a number of ingenious mechanical appliances, things which naturally appeal more to the popular imagination than the subtleties of pure mathematics. ”

“ In the Aristotelian Mechanica we find a remark on the marvel of a great weight being moved by a small force, and the problems discussed bring in the lever in various forms as a means of doing this. We are told also that practically all movements in mechanics reduce to the lever and the principle of the lever (that the weight and the force are in inverse proportion to the distances from the point of suspension or fulcrum of the points at which they act, it being assumed that they act in directions perpendicular to the lever) . ”

“ The sun is hidden from sight, not because it is under the earth, but because it is covered by the higher parts of the earth and because its distance from us is greater. Aristotle adds the detail that the sun is carried round the northern portion of the earth and produces night because the earth is lofty towards the north. We must again conclude that the stars which, like the sun and moon, move laterally round the earth between their setting and rising again are the planets, as distinct from the fixed stars. ”

“ THE WORKS OF ARCHIMEDES.
The range of Archimedes’s writings will be gathered from the list of his various treatises. An extraordinarily large proportion of their contents represents entirely new discoveries of his own. He was no compiler or writer of text-books, and in this respect he differs from Euclid and Apollonius, whose work largely consisted in systematising and generalising the methods used and the results obtained by earlier geometers. There is in Archimedes no mere working-up of existing material; his objective is always something new, some definite addition to the sum of knowledge. ”

“ As there is so perceptible a change of horizon between places so near to each other, it follows not only that the earth is spherical but also that it is not a very large sphere. Aristotle adds that people are not improbably right when they say that the region about the Pillars of Heracles is joined on to India, one sea connecting them. He quotes a result arrived at by the mathematicians of his time, that the circumference of the earth is 400,000 stades. He is clear that the earth is much smaller than some of the stars, but that the moon is smaller than the earth. ”

“ Of propositions attributed to him the most famous is, of course, the theorem that in a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the sides about the right angle (Eucl. I., 47) ; and, seeing that Greek tradition universally credits him 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. ”

“ The outermost revolution is that of the sphere of the fixed stars, which carries round with it the whole universe including the sun, moon and planets; the latter seven bodies, while they are so carried round by the general rotation, have slower revolutions of their own in addition, one inside the other, these revolutions being at different speeds but all in the opposite sense to the general rotation of the universe. The quickest rotation is that of the fixed stars and the universe, which takes place once in about twenty-four hours. ”

“ Seeing in you, as I say, an earnest student, a man of considerable eminence in philosophy and an admirer of mathematical inquiry when it comes your way, I have thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, which, when you see it, will put you in possession of a means whereby you can investigate some of the problems of mathematics by mechanics. ”

“ The sun, moon and stars are all made of fire and (like the earth) they ride on the air because of their breadth. The sun is flat like a leaf. Anaximenes also held that the stars are fastened on a crystal sphere like nails or studs. It seems clear therefore that by the stars which “ride on the air because of their breadth” he meant the planets only. ”

“ 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 half of the volume by which the cylinder exceeds the prism (since we take away more than half of the excess of the area of the circular base over that of the inscribed polygon, as in Euclid XII., 2) . ”

“ THALES.
Thales of Miletus (about 624–547 B.C.) was a man of extraordinary versatility; philosopher, mathematician, astronomer, statesman, engineer, and man of business, he was declared one of the Seven Wise Men in 582–581 B.C. His propensity to star-gazing is attested by the story of his having fallen into a well while watching the stars, insomuch that (as Plato has it) he was rallied by a clever and pretty maidservant from Thrace for being so “eager to know what goes on in the heavens when he could not see what was in front of him, nay at his very feet”. ”

“ 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 the lesser by such a magnitude as is capable, if added continually to itself, of exceeding any magnitude of the same kind as the original magnitudes. This assumption is known as the Axiom or Postulate of Archimedes, though, as he states, it was assumed before his time by those who used the method of exhaustion. ”

“ Now it was much to discover, as Heraclides did, that the earth rotates about its own axis and that Mercury and Venus revolve round the sun like satellites; and it seems a priori incredible that one man should not only have reached, and improved upon, the hypothesis of Tycho Brahe but should also have suggested the Copernican hypothesis. ”

“ Archimedes then passes to the spiral which, he says, presents another sort of problem, having nothing in common with the foregoing. After a definition of the spiral he enunciates the main propositions about it which are to be proved in the treatise. The spiral (now known as the Spiral of Archimedes) is defined as the locus of a point starting from a given point (called the “origin”) on a given straight line and moving along the straight line at uniform speed, while the line itself revolves at uniform speed about the origin as a fixed point. ”

“ Beyond the above particulars of the life of Archimedes, we have nothing but a number of stories which, if perhaps not literally accurate, yet help us to a conception of the personality of the man which we would not willingly have altered. ”

“ The deductive form of proof by the method of exhaustion is apt to obscure not only the way in which the results were arrived at but also the real character of the procedure followed. What Archimedes actually does in certain cases is to perform what are seen, when the analytical equivalents are set down, to be real integrations; this remark applies to his investigation of the areas of a parabolic segment and a spiral respectively, the surface and volume respectively of a sphere and a segment of a sphere, and the volume of any segments of the solids of revolution of the second degree. ”

“ If the solid is heavier than the fluid, it will, if placed in the fluid, descend to the bottom and, if weighed in the fluid, the solid will be lighter than its true weight by the weight of the fluid displaced (Prop. 7) .
The last-mentioned theorem naturally connects itself with the story of the crown made for Hieron. It was suspected that this was not wholly of gold but contained an admixture of silver, and Hieron put to Archimedes the problem of determining the proportions in which the metals were mixed. ”

“ The diameter of the real universe being 10,000 times that of the so-called universe, the final number of grains of sand in the real universe is found to be 1063, which in Archimedes’s terminology is a myriad-myriad units of the eighth order of numbers. ”