The fifteenth is of a circle. Κοὐκλος ἐστὶ σχῆμα ἐπίπεδον, &c. _A circle is a plain figure comprehended by one line which is called the circumference, to which circumference all the straight lines drawn from one of the points within the figure are equal to one another._ This is true. But if a man had never seen the generation of a circle by the motion of a compass or other equivalent means, it would have been hard to persuade him that there was any such figure possible. It had been therefore not amiss first to have let him see that such a figure might be described. Therefore so much of geometry is no part of philosophy, which seeketh the proper passions of all things in the generation of the things themselves. After the fifteenth till the last or thirty-fifth definition, all are most accurate, but the last which is this, _parallel straight lines are those which being in the same plane, though infinitely produced both ways, shall never meet_. Which is less accurate. For how shall a man know that there be straight lines which shall never meet, though both ways infinitely produced? Or how is the definition of parallels, that is, of lines perpetually equidistant, good, wherein the nature of equidistance is not signified? Or if it were signified, why should it not comprehend as well the parallelism of circular and other crooked lines, as of straight, and as well of superficies, as of lines? By parallels is meant equidistant both lines and superficies, and the word is therefore not well defined without defining first equality of distance. And because the distance between two lines or superficies, is the shortest line that can join them, there either ought to be in the definition the _shortest distance_, which is that of the perpendicular and without inclination, or the distance in equal inclination, that is, in equal angles. Therefore if parallels be defined to be those lines or superficies, where the lines drawn from one to another in equal angles be equal, the definition, as to like lines, or like superficies, will be universal and convertible. And if we add to this definition, that the equal angles be drawn not opposite ways, it will be absolute, and convertible in all lines and superficies; and the definition will be this: _parallels are those lines and superficies between which every line drawn, in any angle, is equal to any other line drawn in the same angle the same way_. For by this definition the distance between them will perpetually be equal, and consequently they will never come nearer together, how much, or which way soever they be produced. And the converse of it will be also true, _if two lines, or two superficies be parallel, and a straight line be drawn from one to the other, any other straight line, drawn from one to the other in the same angle, and the same way, will be equal to it_. This is manifestly true, and, most egregious professors, new, at least to you.