Enhance Your Quote Experience
Enjoy ad-free browsing, unlimited collections, and advanced search features with Premium.
The discovery of Hippocrates amounted to the discovery of the fact that from the relation (1)<math>\frac{a}{x} = \frac{x}{y} = \frac{y}{b}</math>it f… - Thomas Little Heath
" "The discovery of Hippocrates amounted to the discovery of the fact that from the relation
(1)<math>\frac{a}{x} = \frac{x}{y} = \frac{y}{b}</math>it follows that<math>(\frac{a}{x})^3 = [\frac{a}{x} \cdot \frac{x}{y} \cdot \frac{y}{b} =] \frac{a}{b}</math>and if <math>a = 2b</math>, [then <math>(\frac{a}{x})^3 = 2</math>, and]<math>a^3 = 2x^3</math>.The equations (1) are equivalent [by reducing to common denominators or cross multiplication] to the three equations
(2)<math>x^2 = ay, y^2 = bx, xy = ab</math>[or equivalently...<math>y = \frac{x^2}{a}, x = \frac{y^2}{b}, y = \frac{ab}{x}</math> ]and the solutions of Menaechmus described by Eutocius amount to the determination of a point as the intersection of the curves represented in a rectangular system of Cartesian coordinates by any two of the equations (2).
Let AO, BO be straight lines placed so as to form a right angle at O, and of length a, b respectively. Produce BO to x and AO to y.
The first solution now consists in drawing a parabola, with vertex O and axis Ox, such that its parameter is equal to BO or b, and a hyperbola with Ox, Oy as asymptotes such that the rectangle under the distances of any point on the curve from Ox, Oy respectively is equal to the rectangle under AO, BO i.e. to ab. If P be the point of intersection of the parabola and hyperbola, and PN, PM be drawn perpendicular to Ox, Oy, i.e. if PN, PM be denoted by y, x, the coordinates of the point P, we shall have <math>\begin{cases}y^2 = b.ON = b.PM = bx\\ and\\ xy = PM.PN = ab\end{cases}</math>whence<math>\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.</math>
In the second solution of Menaechmus we are to draw the parabola described in the first solution and also the parabola whose vertex is O, axis Oy and parameter equal to a. The point P where the two parabolas intersect is given by<math>\begin{cases}y^2 = bx\\x^2 = ay\end{cases}</math>whence, as before,<math>\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.</math>
English
Collect this quote
About Thomas Little Heath
Sir Thomas Little Heath (5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. Heath translated works of Euclid of Alexandria, Apollonius of Perga, Aristarchus of Samos, and Archimedes of Syracuse into English.
Also Known As
Alternative Names:
Thomas Heath (classicist)
•
Thomas L. Heath
•
Sir Thomas Little Heath
Related quotes. More quotes will automatically load as you scroll down, or you can use the load more buttons.
Additional quotes by Thomas Little Heath
The actual writers of Elements of whom we hear were the following. Leon, a little younger than Eudoxus, was the author of a collection of propositions more numerous and more serviceable than those collected by Hippocrates. Theudius of Magnesia, a contemporary of Menæchmus and Dinostratus, "put together the elements admirably, making many partial or limited propositions more general". Theudius's book was no doubt the geometrical text-book of the Academy and that used by Aristotle.
Diophantos lived in a period when the Greek mathematicians of great original power had been succeeded by a number of learned commentators, who confined their investigations within the limits already reached, without attempting to further the development of the science. To this general rule there are two most striking exceptions, in different branches of mathematics, Diophantos and Pappos. These two mathematicians, who would have been an ornament to any age, were destined by fate to live and labour at a time when their work could not check the decay of mathematical learning. There is scarcely a passage in any Greek writer where either of the two is so much as mentioned. The neglect of their works by their countrymen and contemporaries can be explained only by the fact that they were not appreciated or understood. The reason why Diophantos was the earliest of the Greek mathematicians to be forgotten is also probably the reason why he was the last to be re-discovered after the Revival of Learning. The oblivion, in fact, into which his writings and methods fell is due to the circumstance that they were not understood. That being so, we are able to understand why there is so much obscurity concerning his personality and the time at which he lived. Indeed, when we consider how little he was understood, and in consequence how little esteemed, we can only congratulate ourselves that so much of his work has survived to the present day.
PREMIUM FEATURE
Advanced Search Filters
Filter search results by source, date, and more with our premium search tools.
Archytas of Tarentum found the two mean proportionals by a very striking construction in three dimensions, which shows that solid geometry, in the hands of Archytas at least, was already well advanced. The construction was usually called mechanical, which it no doubt was in form, though in reality it was in the highest 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.
Loading...