As a diplomat, Leibniz travelled widely, and in 1672 went to Paris where he mixed in scientific and mathematical circles. He was advised by the Dutch mathematician Huygens to read Pascal's work of 1659 in which he summed infinite series, found derivatives of the trignometric fuctions geometrically and calculated power series for sine and cosine. In 1673 he met several mathematicians in England and bought Barrow's Lectures in Geometry . He also saw a manuscript of Newton's calculus, but at that stage was not capable of understanding it.

Between 1674 and 1676 he made important discoveries, and constructed a calculating machine. He summed infinite series including Sum 1/ n(n+1), Sum 1/ n(n+1)(n+2) etc. using the idea of difference equations . He rewrote Pascal's proof of sin' x = cos x in terms of increment in y /increment in x , and essentially discovered the algorithms for the sum, product and quotient rule.

Let an axis AX and several curves such as VV, WW, YY, ZZ be given, of which the ordinates VX, WX, YX, ZX, perpendicular to the axis, are called v, w, y, z respectively. The segment AX, cut off from the axis is called x. Let the tangents be VB, WC, YD, ZE, intersecting the axis respectively at B, C, D, E. Now some straight line selected arbitrarily is called dx, and the line which is to dx as v (or w, or y, or z) is to XB (or XC, or XD, or XE) is called dv (or dw, or dy, or dz), or the difference of these v (or w, or y, or z). Under these assumptions we have the following rules of the calculus.

If a is a given constant, then da = 0, and d(ax) = a dx. If y = v (that is, if the ordinate of any curve YY is equal to anv corresponding ordinate of the curve V V , then dy = dv.

Now addition and subtraction: if z - y + w + z = v, then d(z - y + w + x) = dv = dz - dy + dw + dx. Multiplication: d(xv) = x dv + v dx, or, setting y = xv, dy = x dv + v dx. It is indifferent whether we take a formula such as xv or its replacing letter such as y. It is to be noted that x and dx are treated in this calculus in the same way as y and dy, 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.

Now division: d (v/y) or (if z = v/y) dz = (+ or - v dy - or + y dv)/yy .
The following should be kept well in mind about the signs. When in the calculus for a letter simply its differential is substituted, then the signs are preserved; for z we write dz, for -z we write -dz, as appears from the previously given rule for addition and subtraction. However, when it comes to an explanation of the values, that is, when the relation of z to x is considered, then we can decide whether dz is a positive quantity or less than zero. When the latter occurs, then the tangent ZE is not directed toward A, but in the opposite direction, down from X. This happens when the ordinates z decrease with increasing x. And since the ordinates v sometimes increase and sometimes decrease, dv will sometimes be positive and sometimes be negative; in the first case the tangent VB is directed toward A, in the latter it is directed in the opposite sense. None of these cases happens in the intermediate position at M, at the moment when v neither increases nor decreases, but is stationary. Then dv = 0, and it does not matter whether the quantity is positive or negative, since + 0 = - 0. At this place v, that is, the ordinate LM, is maximum (or, when the convexity is turned to the axis, minimum), and the tangent to the curve at M is directed neither in the direction from X up to A, to approach the axis, nor down to the other side, but is parallel to the axis. When dv is infinite with respect to dx, then the tangent is perpendicular to the axis, that is, it is the ordinate itself. When dv = dx, then the tangent makes half a right angle with the axis. When with increasing ordinates v its increments or differences dv also increase (that is, when dv is positive, d dv, the difference of the differences, is also positive, and when dv is negative, d dv is also negative), then the curve turns toward the axis its concavity, in the other case its convexity.

Where the increment is maximum or minimum, or where the increments from decreasing turn into increasing, or the opposite, there is a point of inflection. Here concavity and convexity are interchanged, provided the ordinates too do not turn from increasing into decreasing or the opposite, because then the concavity or convexity would remain. However, it is impossible that the increments continue to increase or decrease, but the ordinates turn from increasing into decreasing, or the opposite. Hence a point of inflection occurs when d dv = 0 while neither v nor dv = 0. The problem of finding inflection therefore has not, like that of finding a maximum, two equal roots, but three. This all depends on the correct use of the signs.

Sometimes it is better to use ambiguous signs, as we have done with the division, before it is determined what the precise sign is. When with increasing x, v/y increases (or decreases), then the ambiguous signs in d(v/y) = ([+ or -] v dy [- or +] y dv)/yy must be determined in such a way that this fraction is a positive (or negative) quantity. But [- or +] means the opposite of [+ or -], so that when one is + the other is -. There also may be several ambiguities in the same computation, which I distinguish by parentheses. ..

Powers: dx^a = ax^a-1 dx; for example, dx³ = 3x² dx. d(1/ x^a) = -(a/x^a-1) dx. For example, if w = 1/x³ then dw = -3/x⁴ dx.

Roots. ....

The law for integral powers would have been sufficient to cover the case of fractions as well as roots, for a power becomes a fraction when the exponent is negative, and changes into a root when the exponent is fractional. However, I prefer to draw these conclusions myself rather than relegate their deduction to others, since they are quite general and occur often. In a matter that is already complicated in itself it is preferable to facilitate the operations.

Knowing thus the Algorithm (as I may say) of this calculus, which I call differential 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 of all this will be easy to one who is experienced in these matters and who considers the fact, until now not sufficiently explored, that dx, dy, dv, dw, dz can be taken proportional to the momentary differences, that is, increments or decrements, of the corresponding x, y, v, w, z. To any given equation we can thus write its differential equation. This can be done by simply substituting for each term (that is, any part which through addition or subtraction contributes to the equation) its differential quantity. For any other quantity (not itself a term, but contributing to the formation of the term) we use its differential quantity, to form the differential quantity of the term itself, not by simple substitution, but according to the prescribed Algorithm. The methods published before have no such transition. They mostly use a line such as DX or of similar kind, but not the line dy which is the fourth proportional to DX, DY, dx---something quite confusing. From there they go on removing fractions and irrationals (in which undetermined quantities occur). It is clear that our method also covers transcendental curve's--those that cannot be reduced by algebraic computation, or have no particular degree and thus holds in a most general way without any particular and not always satisfied assumptions.

We have only to keep in mind that to find a tangent means to draw a line that connects 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 the curve. This infinitely small distance can always be expressed by a known differential like dv, or by a relation to it, that is, by some known tangent. In particular, if y were a transcendental quantity, for instance the ordinate of a cycloid, and it entered into a computation in which z, the ordinate of another curve, were determined, and if we desired to know dz or by means of dz the tangent of this latter curve, then we should by all means determine dz by means of dy, since we have the tangent of the cycloid. The tangent to the cycloid itself, if we assume that we do not yet have it, could be found in a similar way from the given property of the tangent to the circle.

I shall now show that the general problem of quadratures can be reduced to the finding of a line that has a given law of tangency, that is, for which the sides of the characteristic triangle have a given mutual relation. Then I shall show how this line can be described by a motion that I have invented. For this purpose I assume for every curve C(C') a double characteristic triangle,

one, TBC, that is assignable, and one, GLC, that is inassignable, and these two are similar. The inassignable triangle consists of the parts GL, LC, with the elements of the coordinates CF, CB as sides, and GC, the element of arc, as the base or hypotenuse. But the assignable triangle TBC consists of the axis, the ordinate, and the tangent, and therefore contains the angle between the direction of the curve (or its tangent) and the axis or base, that is, the inclination of the curve at the given point C. Now let F(H), the region of which the area has to be squared, be enclosed between the curve H(H), the parallel lines FH and (F)(H), and the axis F(F); on that axis let A be a fixed point, and let a line AB, the conjugate axis, be drawn through A perpendicular to AF. We assume that point C lies on HF (continued if necessary); this gives a new curve C(C') with the property that, if from point C to the conjugate axis AB (continued if necessary) both its ordinate CB (equal to AF) and tangent CT are drawn, the part TB of the axis between them is to BC as HF to a constant segment a, or a times BT is equal to the rectangle AFH (circumscribed about the trilinear figure AFHA).

This being established, I claim that the rectangle on a and E(C) (we must.discriminate between the ordinates FC and (F)(C) of the curve) is equal to the region F(H). When therefore I continue line H(H) to A, the trilinear figure A FHA of the figure to be squared is equal to the rectangle with the constant a and the ordinate FC of the squaring curve as sides.

This follows immediately from our calculus. Let AF = y, FH = z, BT = t, and FC = x; then t = zy: a, according to our assumption; on the other hand, t = y dx:dy because of the property of the tangents expressed in our calculus. Hence a dx = z dy and therefore ax = INT z dy = AFHA. Hence the curve C(C') is the quadratrix with respect to the curve H(H)' while the ordinate FC of C(C'), multiplied by the constant a, makes the rectangle equal to the area, or the sum of the ordinates H(H) corresponding to the corresponding abscissas AF. Therefore, since BT: AF = FH: a (by assumption), and the relation of this FH to AF (which expresses the nature of the figure to be squared) is given, the relation of BT to FH or to BC, as well as that of BT to TC, will be given, that is, the relation between the sides of triangle TBC.

Leibniz did not define his differentials dy and dx as infinitesimal quantities, but as line segments from which he could calculate dy/dx as a ratio. However, he needed infinitesimal methods to derive his algorithms. Note that there is a sign ambiguity in his quotient rule, because all his distances are positive.

By 1690, Leibniz had discovered most of the ideas of elementary calculus, including differential equations, but he did not write up a complete treatment of this material, which was first done by L'Hospital (1651--1704) and Jean Bernoulli (1667--1748).

He was bothered by the use of infinitesimals which he justified in two ways: