CONCHOID

, or Conchiles, the name of a curve invented by Nicomedes. It was much used by the ancients in the construction of solid problems, as appears by what Pappus says.

It is thus constructed: AP and BD being two lines intersecting at right angles; from P draw a number of other lines PFDE, &c, on which take always DE = DF = AB or BC; so shall the curve line drawn through all the points E, E, E, be the first conchoid, or that of Nicomedes; and the curve drawn through all the other points F, F, F, is called the second conchoid; though in reality, they are both but parts of the same curve, having the same pole P, and four infinite legs, to which the line DBD is a common asymptote. Fig. 1 Fig. 2 Fig. 3

The inventor, Nicomedes, contrived an instrument for describing his conchoid by a mechanical motion: thus, in the rule AD is a channel or groove cut, so that a smooth nail, firmly fixed in the moveable rule CB, in the point F, may slide freely within it: into the rule EG is fixed another nail at K, for the moveable rule CB to slide upon. If therefore the rule BC be so moved, as that the nail F passes along the canal AD; the style, or point in C, will describe the first conchoid.

To determine the equation of the curve: put AB = BC = DE = DF = a, PB= b, BG = EH = x, and GE = BH = y; then the equation to the first con- | choid will be ; and, changing only the sign of x, as being negative in the other curve, the equation to the 2d conchoid will be .

Of the whole conchoid, expressed by these two equations, or rather one equation only, with different signs, there are three cases or species; as first, when BC is less than BP, the conchoid will be as in fig. 1; when BC is equal to BP, the conchoid will be as in fig. 2; and when BC is greater than BP, the conchoid will be as in fig. 3.

Newton approves of the use of the conchoid for trisecting angles, or finding two mean proportionals, or for constructing other solid problems. Thus, in the Linear Construction of equations, towards the end of his Universal Arithmetic, he says, “The antients at first endeavoured in vain at the trisection of an angle, and the finding of two mean proportionals by a right line and a circle. Afterwards they began to consider several other lines, as the conchoid, the cissoid, and the conic sections, and by some of these to solve these problems.” A gain, “Either therefore the trochoid is not to be admitted at all into geometry, or else, in the construction of problems, it is to be preferred to all lines of a more difficult description: and there is the same reason for other curves; for which reason we approve of the trisections of an angle by a conchoid, which Archimedes in his Lemmas, and Pappus in his Collections, have preferred to the inventions of all others in this case; because we ought either to exclude all lines, besides the circle and right line, out of geometry, or admit them according to the simplicity of their descriptions, in which case the conchoid yields to none, except the circle.” Lastly, “That is arithmetically more simple which is determined by the more simple equations, but that is geometrically more simple which is determined by the more simple drawing of lines; and in geometry, that ought to be reckoned best which is geometrically most simple: wherefore I ought not to be blamed, if, with that prince of mathematicians, Archimedes, and other antients, I make use of the conchoid for the construction of solid problems.”

CONCRETE Numbers are those that are applied to express or denote any particular subject; as 3 men, 2 pounds, &c. Whereas, if nothing be connected with a number, it is taken abstractedly or universally: thus, 4 signifies only an aggregate of 4 units, without any regard to a particular subject, whether men or pounds, or any thing else.