NOTES.

(Referred to Page 81.)

In articles 66 and 75 I have interpreted the negative solutions by the examination of the equation, which they immediately verify, as I had done before, and this method appeared to me always exact, as the object is merely to show, that these solutions have a rational sense, since they resolve questions analagous to the one proposed; but there are often several ways of forming these questions, and the following, which was communicated to me by M. Français, a distinguished geometer, Professor at the School of Artillery of Mayence, seemed to me more simple, than that given in these Elements.

"He thinks, that we ought to leave out of the enunciation of the question of art. 65 the idea of the departure of the couriers, and to suppose them to have been travelling from an indefinite time; the question then would be stated thus. Two couriers travel the same route in the same direction C' A B C (page 72); after they have proceeded, each a certain time, one finds himself in A at the instant that the other is in B; their distance and rate of going are known; it is asked, at what point of the route they will encounter each other ?

This enunciation leads to the same equation, as that of art. 65; but "the continuity of the motion being once established, the negative solution admits of an explanation without the necessity of changing the direction of one of the couriers. Indeed, since their motion does not commence at the points A and B, but both, before arriving at these points, are supposed to have been going in the same manner for an indefinite time from C' toward B, it is easy to conceive, that the courier, who at this point is in advance of the one at A, who travels slower, must at a certain time have been behind him and overtaken him before his arrival at the point A. The sign then indicates (as in the application of Algebra to Geometry) that the distance AR' is to be taken in a direction opposite to AR, which is regarded as positive. The change to be made in the enunciation, to render the negative solution positive, is reduced to supposing, that the two Alg.

34

couriers must have come together before their arrival at the point A, instead of its taking place afterward."

Indeed, when we place the point R' between A and C, instead of putting it between A and B, we find AB = BR'

results the equation y

xa, instead of x

AR', whence -ya, which we

first obtained; and there is no need of changing the sign of c, the

M. Français applies not less happily these considerations to the case of art. 75, by substituting, for the couriers, moveable bodies, subjected to a continued motion commencing from an indefinite time. He enunciates the problem thus ; "Two moveable bodies are carried uniformly in a straight line CB (page 80) one in the direction BC, and the other in the direction CB with given velocities; that, which is carried in the first direction, is found in B, a known number of hours before the other has arrived at A ; it is asked, at what point of the indefinite straight line BC their meeting takes place?

The solution x —— 48mls. implies, that the two moveable bodies met at the point R, before that, which is carried from C towards B, had reached the point A, and that the second, which moves from B toward C, was at the point C, where he is found when the other is at the point A.”

The position assigned to the point R, verifies itself by observing, that there results from it AC BC AB = cd a, instead of a+cd, as first obtained (page 80,) and, conseqently,

-

In this manner there is no change to be made in the direction of the motion; indeed there is a difference in the circumstances of the problem, and as I said before, this proves, that there are several physical questions corresponding to the same mathematical relations. But the enunciations, here given, have the advantage of not breaking the law of continuity, and this is derived from the consideration of lines, which represent in a manner the most simple and general, the circumstances of a change of sign in magnitudes. (See the Elementary Treatise of Trigonometry and Application of Algebra to Geometry.)

(Note referred to Page 185.)

It may be thought, that, in order to discover the roots of any equation of the fourth degree

x2 + px3 + q x2 + rx + s = 0,

it would be sufficient to compare it with the product of article 183, observing to put equal to each other the quantities by which the same power of x is multiplied; and it is in this manner that most elementary writers think to demonstrate, that an equation of any degree whatever is the product of as many simple factors, as there are units in the exponent of its degree. It will be seen by what follows, that the reasoning by which this is attempted to be proved, is defective. We stated the proposition with qualification in article 182, because it is necessary, in order to establish it unconditionally, to show that an equation of whatever degree has a root, real or imaginary, which is not easily done in an elementary work, and which happily is not necessary. Some remarks relating to this subject may be found in the Supplement.

in order to deduce from them the value of the letters, a, b, c, d, the roots of the proposed equation, the calculation would be very complicated, if, in the determination of the unknown quantities, a, b, c, d, we adopt the method of article 78; but if we muitiply the first of the above equations by a3, the second by a2, the third by a, and add these three products to the fourth, member to member, we shall have a2 = p a3 + q a2 + ra + s, from which we derive, by simple transposition,

a1 + pa3 + q a2 +ra + s = 0.

This equation contains only a, but it is entirely similar to the one proposed. The difficulty of obtaining a, therefore, is the same as that of obtaining z.

"Thus," says Castillon (Mém. de Berlin, année 1789) "it is shown in every work on algebra, that an equation of any degree we please, is formed of several simple binomials, but it is not so evident that an equation, formed by the multiplication of several simple binomials, can have such coefficients as we please."

If, instead of multiplying the first three equations in a, b, c, d, by a3, a2, and a, respectively, we multiply them by b3, b2, and b, or by c3, c2, c, or d3, da, d, and add the products to the fourth equation, we shall have in the first

from which it follows, that we are conducted to the same equation in the case of a, in that of b, &c. Indeed the quantities, a, b, c, d, being all disposed in the same manner in each equation, it is not to be supposed that one should be determined by a different operation from that of the others; and, in general, if in the investigation of several unknown quantities, we are obliged to employ for each the same reasonings, the same operations, and the same known quantities, all these quantities will necessarily be roots of the same equation.