186 A History ofMathematics
AX, cut off from the axis is calledx. Let the tangents beVB,WC,YD,ZE, intersecting the axis
respectively atB,C,D,E. Now some straight line selected arbitrarily is calleddx, and the line which
is todxasv(orw,ory,orz)istoXB(orXC,orXD,orXE) is calleddv(ordw,ordy,ordz), or
the difference of thesev(orw,ory,orz). Under these assumptions we have the following rules of
the calculus.
If ais a given constant, thenda=0, andd(ax)= adx...Now additionandsubtraction:if
z−y+w+x=v, thend(z−y+w+x)=dv=dz−dy+dw+dx. Multiplication:d(xv)=xdv+vdx,
or, settingy=xv,dy=xdv+vdx. It is indifferent whether we take a formula such asxvor its
replacing letter such asy. It is to be noted thatxanddxare treated in this calculus in the same
way asyanddy, or any other indeterminate letter with its difference. It is also to be noted that
we cannot always move backward from a differential equation without some caution, something
which we shall discuss elsewhere.- When with increasing ordinatesvits increments or differences also increase (that is, whendv
is positive,ddv, the difference of the differences, is also positive, and whendvis negative,ddvis also
negative), then the curve turns toward the axis itsconcavity, in the other case itsconvexity. - Knowing thus theAlgorithm(as I may say) of this calculus, which I calldifferential calculus, all
other differential equations can be solved by a common method. We can find maxima and minima
as well as tangents without the necessity of removing fractions, irrationals, and other restrictions,
as had to be done according to the methods that have been published hitherto. The demonstration
will be easy to one who is experienced in these matters and who considers the fact, until now not
sufficiently explored, thatdx,dy,dv,dw,dzcan be taken proportional to the momentary differences,
that is, increments or decrements, of the correspondingx,y,v,w,z...We have only to keep in
mind that to find atangentmeans to connect two points of the curve at an infinitely small distance,
or the continued side of a polygon with an infinite number of angles, which for us takes the place of
thecurve. This infinitely small distance can always be expressed by a known differential likedv,or
by a relation to it, that is, by some known tangent.
Appendix C. From thePrincipia
Book I, Proposition 1, Theorem 1. (Reproduced from Fauvel and Gray 12 B.5.)The areas which revolving bodies describe by radii drawn to an immovable centre of force do lie in the same
immovable planes, and are proportional to the times in which they are described.(See Fig. 5.)
For suppose the time to be divided into equal parts, and in the first part of that time let the body
by its innate force describe the right lineAB. In the second part of that time, the same would (by
Law I.), if not hindered, proceed directly toc, along the lineBcequal toAB; so that by the radii
AS,BS,cS, drawn to the centre, the equal areasASB,BSc, would be described. But when the body
is arrived atB, suppose that a centripetal force acts at once with a great impulse, and, turning
aside the body from the right lineBc, compels it afterwards to continue its motion along the right
lineBC.DrawcCparallel toBSmeetingBCinC; and at the end of the second part of the time,
the body (by Cor. I. of the Laws) will be found inC, in the same plane with the triangleASB. Join
SC, and, becauseSBandCcare parallel, the triangleSBCwill be equal to the triangleSBc, and