APOTOME

, the remainder or difference between two lines or quantities which are only commensurable in power. Such is the difference between 1 and √2, | or the difference between the side of the square and its diagonal.

The term is used by Euclid; and a pretty full explanation of such quantities is given in the tenth book of his Elements, where he distinguishes six kinds of apotomes, and shews how to find them all geometrically.

Apotome Prima, is when the greater term is rational, and the difference of the squares of the two is a square number; as the difference 3-√5.

Apotome Secunda, is when the less number is rational, and the square root of the difference of the squares of the two terms, has to the greater term, a ratio expressible in numbers; such is √18-4, because the difference of the fquares 18 and 16 is 2, and √2 is to √18 as √1 to √9 or as 1 to 3.

Apotome Tertia, is when both the terms are irrational, and, as in the second, the square root of the difference of their squares, has to the greater term, a rational ratio: as √24-√18; for the difference of their squares 24 and 18 is 6, and √6 is to √24 as √1 to √4 or as 1 to 2.

Apotome Quarta, is when the greater term is a rational number, and the square root of the difference of the squares of the two terms, has not a rational ratio to it: as 4-√3, where the difference of the squares 16 and 3 is 13, and √13 has not a ratio in numbers to 4.

Apotome Quinta, is when the less term is a rational number, and the square root of the difference of the squares of the two, has not a rational ratio to the greater: as √6-2, where the difference of the squares 6 and 4 is 2, and √2 to √6 or √1 to √3 or 1 to √3 is not a rational ratio.

Apotome Sexta, is where both terms are irrational, and the square root of the difference of their squares has not a rational ratio to the greater: as √6-√2; where the difference of the squares 6 and 2 is 4, and √4 to √6 or 2 to √6, is not a rational ratio.

The doctrine of apotomes, in lines, as delivered by Euclid in the tenth book, is a very curious subject, and has always been much admired and cultivated by all mathematicians who have rightly understood this part of the elements; and therefore Peter Ramus has greatly exposed his judgment by censuring that book. And the first algebraical writers in Europe commonly employed a considerable portion of their works on an algebraical exposition of that book, which led them to the doctrine of surd quantities; as Lucas de Burgo, Cardan, Tartalea, Stifelius, Peletarius, &c, &c. See also Pappus, lib. 4, prop. 3, and the introduc. to lib. 7. And Dr. Wallis's Algebra, pa. 109.

Apotome, in Music

, is the difference between a greater and less semitone, being expressed by the ratio of 128 to 125.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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APOGEE
APOLLODORUS
APOLLONIUS
APONO (Peter de)
APOPHYGE
* APOTOME
APPARENT
APPARITION
APPEARANCE
APPLICATE
APPROACH