Mathematics

6 NEWTON.

Genita and Moments

LEMMA II

The moment of any genitum is equal to the moments of each of the generating sides multiplied by the indices of the powers of those sides, and by their coefficients continually.

I call any quantity a genitum¹ which is not made by addition or subtraction of divers parts, but is generated or produced in arithmetic by the multiplication, division, or extraction of the root of any terms whatsoever; in geometry by the finding of contents and sides, or of the extremes and means of proportionals. Quantities of this kind are products, quotients, roots, rectangles, squares, cubes, square and cubic sides, and the like. These quantities I here consider as variable and indetermined, and increasing or decreasing, as it were, by a continual motion or flux; and I understand their momentary increments or decrements by the name of moments; so that the increments may be esteemed as added or affirmative moments; and the decrements as subtracted or negative ones. But take care not to look upon finite particles as such. Finite particles are not moments, but the very quantities generated by the moments. We are to conceive them as the just nascent principles of finite magnitudes. Nor do we in this Lemma regard the magnitude of the moments, but their first proportion, as nascent. It will be the same thing, if, instead of moments, we use either the velocities of the increments and decrements (which may also be called the motions, mutations, and fluxions of quantities), or any finite quantities proportional to those velocities. The coefficient of any generating side is the quantity which arises by applying the genitum to that side.

Wherefore the sense of the Lemma is, that if the moments of any quantities A, B, C, &c., increasing or decreasing by a continual flux, or the velocities of the mutations which are proportional to them, be called a, b, c, &c., the moment or mutation of the generated rectangle AB will be

the moment of the generated content ABC will be and the moments of the generated powers will be respectively; and, in general, that the moment of any power will be Also, that the moment of the genitum A²B will be the moment of the generated quantity A³B⁴C² will be and the moment of the generated quantity or will be and so on. The Lemma is thus demonstrated.

Case 1. Any rectangle, as AB, augmented by a continual flux, when, as yet, there wanted of the sides A and B half their moments

and was into or but as soon as the sides A and B are augmented by the other half-moments, the rectangle becomes into or From this rectangle subtract the former rectangle, and there will remain the excess Therefore with the whole increments a and b of the sides, the increment of the rectangle is generated. Q.E.D.

Case 2. Suppose AB always equal to G, and then the moment of the content ABC or GC (by Case 1) will be

that is (putting AB and for G and g), And the reasoning is the same for contents under ever so many sides. Q.E.D.

Case 3. Suppose the sides A, B, and C, to be always equal among themselves; and the moment

of A², that is, of the rectangle AB, will be 2aA; and the moment of A³, that is, of the content ABC, will be 3aA². And by the same reasoning the moment of any power Aⁿ is Q.E.D.

Case 4. Therefore since

into A is 1, the moment of multiplied by A, together with multiplied by a, will be the moment of 1, that is, nothing. Therefore the moment of or of is And generally since into Aⁿ is 1, the moment of multiplied by Aⁿ together with into will be nothing. And, therefore, the moment of or wi11 be Q.E.D.

Case 5. And since

into is A, the moment of multiplied by will be a (by Case 3); and, therefore, the moment of will be or And generally, putting equal to B, then A^m will be equal to Bⁿ, and therefore equal to and equal to or and therefore is equal to b, that is, equal to the moment of Q.E.D.

Case 6. Therefore the moment of any genitum A^mBⁿ is the moment of A^m multiplied by Bⁿ, together with the moment of Bⁿ multiplied by A^m, that is,

and that whether the indices m and n of the powers be whole numbers or fractions, affirmative or negative. And the reasoning is the same for higher powers. Q.E.D.

Corollary I. Hence in quantities continually proportional, if one term is given, the moments of the rest of the terms will be aa the same terms multiplied by the number of intervals between them and the given term. Let A, B, C, D, E, F be continually proportional; then if the term C is given, the moments of the rest of the terms will be among themselves as -2A, -B, D, 2E, 3F.²

Corollary II. And if in four proportionals the two means are given, the moments of the extremes will be as those extremes. The same is to be understood of the sides of any given rectangle.

Corollary III. And if the sum or difference of two squares is given,³ the moments of the sides will be inversely as the sides.

¹ A genitum is therefore an expression of one term which is dependent on one variable. The Motte-Cajori translation is genitum, making the term neuter, but in Latin it is quantitas genita, a generated quantity.

² When

and then C is constant and

³ When

then Newton, who in Book I has used for his proofs only his theorem of prime and ultimate ratios, in the sections of Book II following his lemma on genita uses occasionally that part of his fluxional calculus which is expressed by the lemma, that is, the formula Newton’s notation is usually like our own, but occasionally he writes for A² either Aq, or A quad.