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Hippocrates did not indeed solve the problem of duplication, but reduced it to another, namely that of finding two mean proportionals in continued proportion between two given straight lines; and the problem was ever afterwards attacked in this form. Democritus wrote a large number of mathematical treatises, the titles only of which are preserved.
A merchant does not hesitate to multiply the second and third together and divide the product by the first, either because he has not yet forgotten the things which he heard without any demonstration from his school-master, or because he has seen the truth of the rule with the more simple numbers, or because from the 19th Prop. in the 7th book of Euclid he understands the common property of all proportionals.
Having reflected on the singular numerical phenomena of the existence of one mean proportional between two square numbers are rather perhaps only between the two lowest squares; and of two mean proportionals between two cubes, perhaps again confining his attention to the two lowest cubes, he finds in the latter symbol an expression of the relation of the elements, as in the former an image of the combination of two surfaces.
Pythagoras is said to have discovered the theory of proportionals or proportion. where the second and third terms are respectively the arithmetic and harmonic mean between a and b. A particular case is 12:9=8:6.
Between fire and earth, the two extremes, he remarks that there are introduced, not one, but two elements, air and water, which are compared to the two mean proportionals between two cube numbers. The vagueness of his language does not allow us to determine whether anything more than this was intended by him.
For example, would you have him find a mean proportional between two lines, contrive that he should require to find a square equal to a given rectangle; if two mean proportionals are required, you must first contrive to interest him in the doubling of the cube. See how we are gradually approaching the moral ideas which distinguish between good and evil.
The meaning of the words that 'solid bodies are always connected by two middle terms' or mean proportionals has been much disputed. The square of any such number represents a surface, the cube a solid.
Menaechmus, a pupil of Eudoxus, was the discoverer of the conic sections, two of which, the parabola and the hyperbola, he used for solving the problem of the two mean proportionals. If a:x=x:y=y:b, then x²=ay, y²=bx and xy=ab.
Let this point of contact be at I, then making KC, QC, DC proportionals, draw DI parallel to CM; also join CI. I say that CI will be the required refraction of the ray RC. This will be manifest if, in considering CO, which is perpendicular to the ray RC, as a portion of the wave of light, we can demonstrate that the continuation of its piece C will be found in the crystal at I, when O has arrived at K.
In geometry he gave the first solution of the problem of the two mean proportionals, using a wonderful construction in three dimensions which determined a certain point as the intersection of three surfaces, a certain cone, a half-cylinder, an anchor-ring or tore with inner diameter nil.
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