Thomas Little Heath Quotes

Hippocrates also attacked the problem of doubling the cube. ...Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in continued proportion between two given straight lines, i.e. finding x, y such that a:x=x:y=y:b, where a, b are the two given straight lines. It is easy to see that, if a:x=x:y=y:b, then b/a = (x/a)³, and, as a particular case, if b=2a, x³=2a³, so that the side of the cube which is double of the cube of side a is found.

An edition is... still wanted which shall, while in some places adhering... to the original text, at the same time be so entirely remodelled by the aid of accepted modern notation as to be thoroughly readable by any competent mathematician, and this want it is the object of the present work to supply.

The researches of the last thirty or forty years into the history of mathematics (I need only mention such names as those of [Carl Anton] Bretschneider, Hankel, Moritz Cantor, [Friedrich] Hultsch, Paul Tannery, Zeuthen, Loria, and Heiberg) have put the whole subject upon a different plane. I have endeavoured in this edition to take account of all the main results of these researches up to the present date. Thus, so far as the geometrical Books are concerned, my notes are intended to form a sort of dictionary of the history of elementary geometry, arranged according to subjects; while the notes on the arithmetical Books VII.-IX. and on Book X follow the same plan.

Greek mathematics reveals an important aspect of the Greek genius of which the student of Greek culture is apt to lose sight.

Euclid's work will live long after all the text books of the present day are superseded and forgotten. It is one of the noblest monuments of antiquity; no mathematician worthy of the name can afford not to know Euclid, the real Euclid as distinct from any revised or rewritten versions which will serve for schoolboys or engineers. And, to know Euclid, it is necessary to know his language, and, so far as it can be traced, the history of the "elements" which he collected in his immortal work.

Theodorus of Cyrene and Theaetetus generalised the theory of irrationals, and we may safely conclude that a great part of the substance of Euclid's Book X. (on irrationals) was due to Theætetus. Theætetus also wrote on the five regular solids, and Euclid was therefore no doubt equally indebted to Theætetus for the contents of his Book XIII. In the matter of Book XII. Eudoxus was the pioneer. These facts are confirmed by the remark of Proclus that Euclid, in compiling his Elements, collected many of the theorems of Eudoxus, perfected many others by Theætetus, and brought to irrefragable demonstration the propositions which had only been somewhat loosely proved by his predecessors.

It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. ...Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4=4x+20 as ᾰτοπος because it would give x=-4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.

In geometry the following theorems are attributed to him [Thales]—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 is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground (this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids).

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.

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>

The work Was begun in 1913, but the bulk of it was written, as a distraction, during the first three years of the war, the hideous course of which seemed day by day to enforce the profound truth conveyed in the answer of Plato to the Delians. When they consulted him on the problem set them by the Oracle, namely that of duplicating the cube, he replied, 'It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another'.
Truly,Greece and her foundations are
Built below the tide of war,
Based on the crystàlline sea
Of thought and its eternity.

The method of exhaustion was not discovered all at once; we find traces of gropings after such a method before it was actually evolved. It was perhaps Antiphon. the sophist, of Athens, a contemporary of Socrates, who took the first step. He inscribed a square (or, according to another account, a 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 maintained that, by continuing it, we should at last arrive at a polygon with sides so small as to make the polygon coincident with the circle. Thought this was formally incorrect, it nevertheless contained the germ of the method of exhaustion.

There is perhaps no question that occupies, comparatively, a larger space in the history of Greek geometry than the problem of the Doubling of the Cube. The tradition concerning its origin is given in a letter from Eratosthenes of Cyrene to King Ptolemy Euergetes quoted by Eutocius...
"Eratosthenes to King Ptolemy greeting.
"There is a story that one of the old tragedians represented Minos as wishing to erect a tomb for Glaucus and as saying, when he heard that it was a hundred feet every way,Too small thy plan to bound a royal tomb.
Let it be double; yet of its fair form
Fail not, but haste to double every side.But he was clearly in error; for when the aides are doubled, the area becomes four times as great, and the solid content eight times as great. Geometers also continued to investigate the question in what manner one might double a given solid while it remained in the same form.

It is a defect in the existing histories that, while they state generally the contents of, and the main propositions proved in, the great treatises of Archimedes and Apollonius, they make little attempt to describe the procedure by which the results are obtained. I have therefore taken pains, in the most significant cases, to show the course of the argument in sufficient detail to enable a competent mathematician to grasp the method used and to apply it, if he will, to other similar investigations.

Archimedes is said to have requested his friends and relatives to place upon his tomb a representation of a cylinder circumscribing a sphere within it, together with the inscription giving the ratio (3/2) which the cylinder bears to the sphere; from which we may infer that he himself regarded the discovery of this ration as his greatest achievement.