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addressed to Dositheus Apollonius Apollonius of Perga Archimedes finds Archimedes proves Archimedes's Archytas Aristarchus Aristarchus of Samos arithmetical centre of gravity circumscribed figures Conoids and Spheroids contains cube Democritus diameter or axis discovered discovery double the number earth elements ellipse equal height Eratosthenes Eucl Euclid Euclid XII Euclid's Book Eudoxus find the centre fluid geometrical proof given straight line grains of sand greater Hieron Hippocrates Hippocrates of Chios inscribed and circumscribed inscribed triangle investigation large numbers lemma lever magnitudes main propositions Marcellus mathematical measured mechanical method of exhaustion moon motion number of grains number of sides parabola parabolic segment parallel parallelogram Plane Equilibriums Plutarch preface addressed prism problem Prop pyramid Pythagoras Pythagoreans regular polygon respectively right-angled cone right-angled triangle Samos Sandreckoner says so-called universe solid spiral square stadia Suppose surface and volume Syracuse Thales Theaetetus Theodorus of Cyrene theorem theory of proportion tion treatise unequal weight
Page 46 - universe' is the name given by most astronomers to the sphere whose centre is the centre of the earth and whose radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account, as you have heard from astronomers.
Page 46 - His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.
Page 2 - ... not deign to leave behind him any written work on such subjects, but, regarding as ignoble and sordid the business of mechanics and every sort of art which is directed to use and profit, he placed his whole ambition in those speculations in whose beauty and subtlety there is no admixture of the common needs of life J.
Page 32 - First then I will set out the very first theorem which became known to me by means of mechanics, namely, that Any segment of a section of a right-angled cone (ie, a parabola) is fourthirds of the triangle which has the same base and equal height, and after this I will give each of the other theorems investigated by the same method. Then, at the end of the book, I will give the geometrical [proofs of the propositions] .... [I premise the following propositions which I shall use in the course of the...
Page 32 - I deem it necessary to expound the method partly because I have already spoken of it and I do not want to be thought to have uttered vain words, but equally because I am persuaded that it will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me.
Page 32 - I am persuaded, no less useful even for the proof of the theorems themselves ; 'for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.
Page 11 - In a right.angled triangle, the square on the hypotenuse is equal to the sum of the squares on the sides containing the right angle . . . . 130 Applications of Pythagoras' theorem . . . . 132 THEOREM 6.
Page 7 - Geometry is said by many to have been invented among the Egyptians, its origin being due to the measurement of plots of land. This was necessary there because of the rising of the Nile, which obliterated the boundaries appertaining to separate owners.
Page 46 - But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the earth filled up in the way described, but also that of a mass equal in magnitude to the universe. Now you are aware that
Page 32 - ... method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge. This is a reason why, in the case of the theorems the proof of which Eudoxus was the first to discover, namely that the cone is a third part of the cylinder, and the pyramid of the prism, having the same base and equal height, we should give no small share of the credit...