The opinion is currently held among mathematicians that analytic geometry sprang full-armed from the head of DESCARTES, as did Athene from that of Jove. " Proles sine matre creata" is CHASLES' picturesque way of saying the same thing. There is much to be said in favor of this thesis, but the question is too important to be dismissed in four words, even in four Latin words, and another opinion is certainly possible. The fact is that in inquiring into the origin of analytic geometry we run into that difficulty which lies at the bottom of a good proportion of our disputes in this vale of tears. What do we mean by our principal terms? Exactly what do we mean by the words analytic geometry ? Till that is settled, it is futile to ask who discovered it. A very definite view on this important question was expressed by GUNTHER. (i) He based his reasoning on the axiom that there are three stages in the development of analytic geometry and anything which has not passed through these three stages does not deserve the name. a. The determination of points by coordinates. b. The representation of a law by graphing the dependent variable against the independent. c. The determination of the corresponding equation. Now, says GUNTHER, wherever we find an example of the third of these without the first two, we may recognize a stroke of genius on the part of the author but not an anticipation of analytic geometry. This opinion was very upsetting to the ever-kindly ZEUTHEN, who held strongly to the thesis, which I shall presently defend,