Euclid's Elements

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and their order, let it be said briefly. All mathematics revolves around quantity and, with its help, accomplishes whatever it undertakes. Hence, it is easy to know how many and what the parts of this discipline are. For who is unaware that quantity is either continuous or discrete? And that each of these is divided in two ways, because the continuous is mutable and immutable, while the discrete is considered either by itself or in relation to something, such that there is a fourfold type of mathematics. Therefore, the science that contemplates continuous magnitudes and figures that are not mobile claims for itself the name of Geometry. + It is the science of continuous quantity that is immobile in position. That which contemplates mobile and continuous quantity is called Astronomy. It is the knowledge of continuous quantity that is always mobile and of the things that occur through its motion. In the same way, Arithmetic obtains discrete quantity, which considers number, whether even or odd, not by comparing it to another, but by itself. It is the science of discrete quantity, known by itself. Music revolves around the mutual relationship of sounds, from which harmony is formed, because of quantity that is discrete but joined in another way. It is the knowledge of discrete quantity compared to one another and in relation to something else.

But before we enumerate the other species of mathematics, we must explain the reason and reveal the mode by which we say that the mathematician is subject to continuous and discrete quantity, as authorized by the learned. For this is not to be understood absolutely regarding the "how many," which is in sensible things, nor the "how much," which is conceived regarding bodies; for this contemplation is contained by the boundaries of physicists rather than mathematicians. Therefore, of the things that are in a natural body and cannot be separated from it, some can be removed neither in reality nor in thought—such as heat, cold, and dryness—because the body obtains these insofar as it is natural. Others, even if they cannot be separated in reality, we nevertheless imagine in our minds to be absent, because the body does not have these per se, nor insofar as it is endowed with nature, but per accidens by accident, such as straight, curved, bent, and others of that kind. Therefore, the mathematician revolves around quantity and forms that are separable from matter in this way: ἐκ τῆς ἀφαιρέσεως by abstraction. And he provides their definitions without touching upon matter. What is a line? μῆκος ἀπλατὲς a length without breadth. What is a triangle? A figure that is contained by three straight lines. And a circle is a figure that is comprehended by one line. There is no mention of matter here, no trace of it, for the reason just brought forward. Yet let no one suspect that mathematical sciences fall into any error because they rely on so weak and feeble a subject, which is possessed only by thought. For the geometrician uses imagination as if it were an abacus, dividing magnitudes, measuring intervals, and describing lines. Yet all these are not mere figments, but certain forces that have some connection with nature; they cannot be called mere dreams. Nor are the mathematical disciplines contaminated by any lie through the imagination of these things. Just as they differ from Divine [substances] by the condition of the subject matter, so they far excel them by the constant and certain demonstration of reason.

But let us now recount the remaining species of mathematics. Therefore, having made another division, we say that the mathematical faculty revolves either in things purely intellectual or in sensible things. We call those things intellectual which the soul itself excites by itself, separating itself from material forms. And indeed, we place two principal and far-outstanding species of this kind: Arithmetic and Geometry. But of that kind which exercises its office and work in sensible things, six parts are usually made: Mechanics, Astrology, Optics, Geodesy, Canonics or Music, and Computing. Geometry is again divided into the contemplation of planes and solids, which is called stereometry, provided that there is no peculiar treatment of points and lines, since no figure could exist in these without planes or solids. For Geometry does nothing else anywhere but either construct planes and solids, or compare them, once constructed, or divide them. Arithmetic is similarly divided into the contemplation of linear, planar, and solid numbers; for it considers the species of number proceeding from unity, and the origin of planar numbers, both similar and dissimilar, and the progression up to the third multiplication. Geodesy and Computing correspond to these, not treating of intellectual numbers or figures, but of sensible ones. For it does not pertain to Geodesy to measure a cylinder or a cone, but heaps, as it measures cones, and pits, as cylinders; and it does not accomplish this with intellectual straight lines, but with sensible ones—sometimes indeed with things somewhat more certain, like solar rays, and sometimes with coarser things, like ropes and a plumb line. Nor does the computer consider the passions of numbers in themselves, but as they are involved in sensible things. Again, Optics and Canonics have their origin from Geometry and Arithmetic. For Optics indeed uses visual rays as lines and the angles that consist of them. It is divided into three parts.

A small ink sketch at the bottom right corner resembles a human facial profile.