DETENTS

, in a clock, are those stops which, by being lifted up or let down, lock and unlock the clock in striking.

Detent-Wheel, or Hoop-Wheel, that wheel in a clock which has a hoop almost round it, in which there is a vacancy, where the clock locks.

DETERMINATE Number. See Number.

Determinate Problem, is that which has but one solution, or a certain limited number of solutions; in contradistinction to an indeterminate problem, which admits of infinite solutions.

Such, for instance, is the problem, To form an isosceles triangle on a given line, so that each of the angles at the base shall be double of that at the ver- tex; which has only one solution: or this, To find an isosceles triangle whose area and perimeter are given; which admits of two solutions.

Determinate Section, the name of a Tract, or General Problem, written by the ancient geometrician Apollonius. None of this work has come down to us, excepting some extracts and an account of it by Pappus, in the preface to the 7th book of his Mathematical Collections. He there says that the general problem was, “To cut an infinite right line in one point so, that, of the segments contained between the point of section sought, and given points in the said line, either the square on one of them, or the rectangle contained by two of them, may have a given ratio, either to the rectangle contained by one of them and a given line, or to the rectangle contained by two of them.”

Pappus farther informs us, that this Tract of Apollonius was divided into two books; that the first book contained 6 problems, and the second 3; that the 6 problems of the first book contained 16 epitagmas, or cases, respecting the dispositions of the points; and the second book 9. Farther, that of the epitagmas of the 6 problems of the first book, 4 were maxima, and one a minimum: that the maxima are at the 2d epitagma of the 2d problem, at the 3d of the 4th, the 3d of the 5th, and the 3d of the 6th; but that the minimum was at the 3d epitagma of the 3d problem. Also, that the second book eontained three determinations; of which the 3d epitagma of the 1st problem, and the 3d of the 2d were minima, and the 3d of the 3d a maximum. Moreover, that the first book had 27 lemmas, and the second book 24; and lastly, that both books contained 83 theorems.

From such account of the contents of this Tract, and the lemmas also given by Pappus, several persons have attempted to restore, or recompose what they thought might be nearly the form of Apollonius's tract, or the subject of each problem, case, determination, &c; among whom are, Snellius, an eminent Dutch mathematician of the last century; a translation of whose work was published in English by Mr. John Lawson, in 1772, together with a new restoration of the whole work by his friend Mr. William Wales.