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in general, because it is possible (1) for the two points B and Bj
to coincide at a single point, which will happen when the circle
described with the centre A and radius AB touches the straight line PB; and (2) that the circle may not cut the straight line PB at all, in which case the rest of the construction is impossible,
and the same is also to be said regarding the points C, D. Ac- cordingly, drawing the line GF parallel to BP and at a distance from it equal to the given line AB, the point F at which this line cuts the line PE, prolonged if necessary, will be the limit beyond which the points A must not be taken if we desire to obtain possible solutions. There exist also limits for the points B and C, which may be employed in restricting the primitive suppositions made with respect to the distance PA.
The eight points D, which depend in general on each point A, answer to the eight solutions of which the problem is susceptible, and when one has no special datum by means of which it can be determined which of these solutions answer best to the case proposed, it is indispensable to ascertain them all by employing for each one of the eight combinations a special curve of errors. But if it be known, for example, that the distance of the observer to the second object is greater or less than his distance to the
first, it will then be necessary to take on the line PB only the point B in the first case and the point Bj in the second,—a course which will reduce the eight combinations one-half. If we
had the same datum with regard to the third object relatively to the second, and with regard to the first object relatively to the third, then the points C and D would be determined, and we should have but a single solution.
These two examples may suffice to illustrate the uses to which the method of curves can be put in solving problems. But this method, which we have presented, so to speak, in a mechanical manner, can also be submitted to analysis.
The entire question in fact is reducible to the description of a curve which shall pass through a certain number of points, whether these points be given by calculation or construction,
or whether they be given by observation or single experiences entirely independent of one another. The problem is in truth indeterminate, for strictly speaking there can be made to pass through a given number of points an infinite number of differ- ent curves, regular or irregular, that is, subject to equations or arbitrarily drawn by the hand. But the question is not to find any solutions whatever but the simplest and easiest in practice.
Thus if there are only two points given, the simplest solution is a straight line between the two points. If there are three points given, the arc of a circle is drawn through these points, for the arc of a circle after the straight line is the simplest line that can be described.
But if the circle is the simplest curve with respect to de- scription, it is not so with respect to the equation between its abscissæ and rectangular ordinates. In this latter point of view, those curves may be regarded as the simplest of which the or- dinates are expressed by an integral rational function of the abscissæ, as in the following equation
y = a + bx + cx2 + dx3 + . . . ,
where y is the ordinate and x the abscissa. Curves of this class are called in general parabolic, because they may be regarded as a generalisation of the parabola,—a curve represented by the fore- going equation when it has only the first three terms. We have already illustrated their employment in resolving equations, and their consideration is always useful in the approximate descrip- tion of curves, for the reason that a curve of this kind can always be made to pass through as many points of a given curve as we please,—it being only necessary to take as many undetermined coefficients a, b, c, . . . as there are points given, and to determine
these coefficients so as to obtain the abscissæ and ordinates for these points. Now it is clear that whatever be the curve pro- posed, the parabolic curve so described will always differ from it by less and less according as the number of the different points is larger and larger and their distance from one another smaller and smaller.
Newton was the first to propose this problem. The following is the solution which he gave of it:
Let P , Q, R, S, . . . be the values of the ordinates y corre- sponding to the values p, q, r, s, . . . of the abscissæ x; we shall have the following equations
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