# History of the derivative/tangent of a curve

I just want to know the history of the derivative. Whenever I Google for it, I find the history of calculus or the tangent of a curve. However, they barely touch upon what happened before Leibniz and Newton.

One thing I cannot find is: How was the tangent of a curve defined before the derivative? I am looking for a mathematical definition.

• This well-known quote of Felix Klein seems worth putting here: Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions. – Dave L Renfro Jul 18 '16 at 15:32
• – Ben Crowell Jul 20 '16 at 1:58
• Often in old books they recognize tangent lines using "double points". In the equation you solve to find where the line intersects the curve, if there is a root of multiplicity 2 (or more), so that the intersection is a double (or more) point, then it is a tangent line. – Gerald Edgar Jul 20 '16 at 15:24

Keep in mind that the repertoire of curves available before calculus was very limited. The earliest known definition is given for circles by Euclid in book III of Elements:"A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle". In other words, the tangent is a line that meets a curve but stays "on one side" of it.

This may seem narrow, but it works for convex curves like conic sections that Greeks focused on, and even the Archimedean spiral. It was good enough for Euclid to demonstrate Proposition III.16 whose Corollary states:"From this it is clear that the straight line drawn at right angles to the diameter of a circle from its end touches the circle". The proposition itself, and reasoning in its demonstration, are quite interesting:

The straight line drawn at right angles to the diameter of a circle from its end will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed, further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle.

What Euclid says is that the angle between a circle and its tangent is smaller than any rectilinear angle, i.e. it is "infinitesimal" relative to them. Such angles came to be called horn angles, and they are the first example of magnitudes violating the so-called Archimedean axiom (of Eudoxus), and Euclid's own treatment of "magnitudes" as always having ratios to each other. Horn angles and their relation to tangency were widely discussed during Renaissance and in 17th century, motivating some calculus ideas.

By the time of Archimedes Greeks already knew how to draw tangents to all conic sections, and Apollonius's demonstrations in Conica are along the lines of Elements III.16. But "touches" is good enough informally as long as one wants to just find tangents without defining them. Archimedes possibly first used kinematic reasoning to find tangents to his spiral, and the idea was developed in 17th century, see Who discovered the power rule for derivatives? Descartes and others also had an algebraic way of finding tangents before Newton and Leibniz, see Is there a 'lost calculus'?

Carl B. Boyer's The History of the Calculus and Its Conceptual Development is especially detailed on early work relating to calculus (it pretty much fizzles out in the mid 1800s, though), and for curves you'll also want to look at Boyer's History of Analytic Geometry.

• This would be better as a comment rather than an answer. – Ben Crowell Aug 9 '16 at 17:19

The study of tangents goes back at least to Apollonius but was given a strong impetus through the work of Pierre de Fermat. Fermat developed a technique called adequality or "approximate equality" which would amount in modern terminology to dropping second-order terms in an equation involving a small $E$. Fermat did not have the notion of derivative but he made great strides toward developing the techniques that would later be unified by Leibniz and Newton. For a recent discussion of his work at MO see https://mathoverflow.net/questions/247065/fermats-enemies.