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ANALYTICAL GEOMETRY 39 symbolism say, not as operations with operations with integers, but as other operations with integers, they express true propositions. Thus (a + 6) 2 = a 2 + 2a6 + 6 2 expresses, for example, a relation holding between those operations with integers that we call "fractionary numbers," or an analogous relation between integers. The language of algebra is a wonderful instrument for expressing shortly, perspicuously, and suggestively, the exceedingly complicated relations in which abstract things stand to one another. The motive for studying such relations was originally, and is still in many cases, the close analogy of relations between certain abstract things to relations between certain things we see, hear, and touch in the world of actuality round us, and our minds are helped in discovering such analogies by the beautiful picture of alge- braical processes made in space of two or of three dimensions made by the " analytical geometry " of Descartes, described in the next chapter. CHAPTER III THE RISE AND PROGRESS OF MODERN MATHEMATICS ANA- LYTICAL GEOMETRY AND THE METHOD OF INDIVISIBLES WE will now return to the consideration of the first two sections of Descartes' book Geometry of 1637. In Descartes' book we have to glean here and there what we now recognise as the essential points in his new method of treating geometrical questions. These points were not ex- pressly stated by him. I shall, however, try to state them in a small compass. Imagine a curve drawn on a plane surface. This curve may be considered as a picture of an algebraical equation involving x and y in the following way. Choose any point on the curve : and call " x " and " y " the numbers that express the perpendicular distances of this point, in terms of a unit of length, from two straight lines (called " axes ") drawn at right angles to one another in the plane mentioned. Now, as we move from point to point of the curve, x and y both vary, bid there is an unvarying relation which connects x and y, and this relation can be expressed by an algebraical equation
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