, the outside of an arch of a bridge, vault, &c.

EXTREME-and-Mean Proportion, is when a line, or any quantity is so divided, as that the whole line is to the greater part, as that greater part is to the less part. Hence, in any line so divided, the rectangle of the whole line and the less segment, is equal to the square of the greater segment.

Euclid shews how to divide a line in Extreme-andmean ratio, in his Elements, book 2, prop. 11, to this effect: Let AB be the given line; to which draw AE perpendicular and equal to half AB; in EA produced take EF = EB, so shall AF be equal to the greater part; consequently if AG be taken equal to AF, the line AB will be divided in G as required.

The same may be done otherwise thus:

As before, make AE (fig. 2.) perpendicular and = (1/2)AB; join EB, on which take EC = EA, and then take BD = BC, so shall the line be divided in D as required.

No number can be divided into extreme and mean proportion, so that its two parts shall be rational; as is well demonstrated by Clavius, in his Commentary upon the 9th book of Euclid's Elements; and the same thing will also appear from the following algebraical solution of the same problem: Let a denote the whole line, and x the greater part; then shall a-x be the less part, and | the rectangle of the whole and less part being put equal to the square of the greater part, gives this equation, ; hence and by completing the square, and extracting the root, &c, there is at last the greater part; consequently is the less part. And as the square root of 5, which cannot be exactly extracted, makes a portion of both these parts, it is manifest that neither of them can be obtained in rational numbers.

Euclid makes great use of this problem, viz, in several parts of the 13th book of the Elements; and by means of it he constructs that notable proposition, viz the 10th of the 4th book, which is to construct an isosceles triangle having each angle at the base double the angle at the vertex.

EXTREMES Conjunct, and Extremes Disjunct, in Spherical Trigonometry, are the former the two circular parts that lie next the assumed middle part, and the latter are the two that lie remote from the middle part. These were terms applied by lord Napier, in his universal theorem for resolving all right-angled and quadrantal spherical triangles, and published in his Logarithmorum Canonis Descriptio, an. 1614. In this theorem, Napier condenses into one rule, in two parts, the rules for all the cases of right-angled spherical triangles, which had been separately demonstrated by Pitiscus, Lansbergius, Copernicus, Regiomontanus, and others. In this theorem, neglecting the right angle, Napier calls the other five parts, circular parts, which are, the two legs about the right angle, and the complements of the other three, viz of the hypothenufe, and the two oblique angles. Then, taking any three of these five parts, one of them will be in the middle between the other two, and these two are the Extremes Conjunct when they are immediately adjacent to that middle part, or they are the Extremes Disjunct when they are each separated from the middle one by another part. Thus, the five parts being AB, AC, and the complements of BC and of the two angles B and C: then if the three parts be AB, and the complements of the angle B and hypothenuse BC be taken, these three are contiguous to each other, the angle B lying in the middle between the other two; therefore the comp. of B is middle part, and AB with the comp. of BC the Extremes Conjunct. But if the three sides be taken; BC is equally separated from the two legs AB and AC, by two angles B and C; and therefore these two legs AB and AC are Extremes Disjunct, and the comp. of BC the middle part.

Napier's rule for resolving each case is in two parts, as below:

The rectangle contained by radius and the sine of the middle part, is equal to the rectangle of the tangents of the Extremes conjunct, or equal to the rectangle of the fines of the Extremes disjunct. Which rule comprehends all the cases that can happen in right-angled spherical triangles; in the application of which rule, the equal rectangles are divided into a proportion or analogy, in such manner that the term sought may be the last of the four terms that are concerned, and consequently i<*>s corresponding term in the same rectangle must be the first of those terms.

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

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