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according actually appear applied Aristotle arithmetic assumed astronomy attributed base beginning Book called centre circle clear construction contained corresponding course cube curve definition described diameter discovered discovery divided division draw Elements equal equation Eucl Euclid Eudoxus expressed fact figure follows four geometry given gives greater Greek important included irrational kind known length less letters magnitudes mathematics means measure mentioned method motion move multiple namely nature object observed original Pappus parallel particular passage plane Plato position possible probably problem Proclus produced proof proportion propositions proved pyramid Pythagoras Pythagoreans ratio rectangle reference regarded relation represented respectively right angles says seems seen segments sense side similar solid solution sphere square straight line suppose theorem theory things third tion translation triangle true units whole
Page 162 - Thus when it is said that the sum of the three angles of any triangle is equal to two right angles, this is a theorem, the truth of which is demonstrated by Geometry.
Page 381 - Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
Page 63 - ... they saw that the modifications and the ratios of the musical scales were expressible in numbers; — since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number.
Page 8 - Hence when all such inventions were already established, the sciences which do not aim at giving pleasure or at the necessities of life were discovered, and first in the places where men first began to have leisure. This is why the mathematical arts were founded in Egypt; for there the priestly caste was allowed to be at leisure.
Page 377 - If from a point within a circle more than two equal straight lines can be drawn to the circumference, that point is the centre of the circle.
Page 374 - After remarking that the mathematician positively knows that the sum of the three angles of a triangle is equal to two right angles...
Page 198 - Y3 inscribed regular figures of sixteen sides, &r. the preceding process gives the proof that circles are to one another as the squares on their diameters.
Page ix - But Greece and her foundations are Built below the tide of war, Based on the crystalline sea Of thought and its eternity; Her citizens, imperial spirits, Rule the present from the past, On all this world of men inherits Their seal is set.
Page 381 - Heib., vol. v, p. 282. it shows that the magnitudes must be of the same kind, but because, while it includes incommensurable as well as commensurable magnitudes, it excludes the relation of a finite magnitude to a magnitude of the same kind which is either infinitely great or infinitely small ; it is also practically equivalent to the principle which underlies the method of exhaustion now known as the Axiom of Archimedes. Most important of all is the fundamental definition (5) of magnitudes which...