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HISTORICAL VIEW OF THE

are three co-ordinates concerned, and they are connectedby two equations, each involving some two of thevariables.

We have hitherto supposed the equations employedto be what are distinctively called algebraic; that is,involving only simple, finite, algebraical quantities asthe variables. But other classes of equations, which donot fall under these limitations, may yet have a similargeometrical application. For instance, if the variablesdepend upon one another in some relation assigned bytrigonometry, or by logarithms, they may still havecorresponding loci, provided those relations can be ac-tually exhibited. Such are called transcendental curves.Simple cases of these are found in what are called the tri-gonometrical curves, which were invented by E. Wright,about 16'00. The curve of sines is constructed by takingequal distances or abscisses along one axis correspondingto arcs, whilst the ordinates are the sines of those arcs.Their summits trace out the locus, otherwise called thesinusoid. In like manner we have the curve of tan-gents, of cosines, &c. &c. Again, the logarithmic curve.will, from its name, be readily understood as constructedupon an analogous principle. It was invented by JamesGregory.

Most of these and other curves formed subjects ofconsiderable discussion at the period of their invention,the interest of which has in a great degree gone by.Nevertheless, they were of essential value to the pro-gress of discovery, as furnishing the tests by whichthose more general principles of analysis were tried,which were now beginning to be discovered.

Infinitesimals.

We have before noticed the advances made by Keplerin that part of the science of quantity which dependsupon the extension of the ancient method of limits.Galileo , in a very curious passage in the " Dialogues onMotion,” introduces a discussion in which similar prin-