Euclid's Elements

in Optics, which has obtained a name of its own, Catoptrics, and Scenography. Optics provides the causes of those things which appear to us differently than they truly are, due to the varying positions and distances of the objects viewed. Catoptrics is concerned with various and manifold reflections, and is involved with conjectural knowledge. Scenography shows how those things which appear in images do not seem inconsistent or deformed, according to the distances and heights of the objects being designed. Therefore, it does not prescribe the imitation of true equality and proportion, but rather that which strikes our vision in a seemly and appropriate manner; thus, when circles are to be represented, sometimes not circles, but ellipses are described, and squares are made longer on one side. Canonic, or Music, considers the apparent proportions of harmonies, finding the divisions of the rules, and using the senses everywhere as an aid. Mechanics is concerned with sensible things and those joined to matter, while it either prepares instruments of war, such as those Archimedes devised when Marcellus was pressing Syracuse with a heavy siege, or constructs certain admirable things with the greatest artifice using spirits, weights, and ropes, such as Ctesibius, Hero, and Archimedes proposed to be viewed by the men of their times, not without the greatest astonishment. For who would not admire, to omit other things, that glass sphere of Archimedes? And why would one not venerate the mathematical faculties, which can perform such things, to the highest degree?

The zodiacal sign, having feigned the year, runs its course,
And the moon returns, simulated, in a new month.

Thus, as Jupiter elegantly exclaims in Claudian:

Has the power of mortal care progressed to this?
Now my labor is played with in fragile art.

What of the fact that they say Archytas was so capable in this matter that he made a wooden dove flying in the air, as if it were endowed with a soul and supporting itself? Astrology discourses on worldly motions, on the size and shape of celestial bodies, and on their power of illumination, as well as on their distance from us. Its parts are Gnomonics, Meteorscopics, and Dioptrics. Gnomonics is concerned with the measurement of hours through the positions of gnomons, of which Ptolemy treats at length in the book inscribed "On the Analemma." Meteorscopics investigates the differences in elevations and the distances of the stars, and teaches many other various theorems that pertain to Astrology. Dioptrics investigates the distances of the sun, the moon, and other stars through instruments of this kind. But let it be enough to have spoken of these things briefly thus far. But since many, especially in these times, are excited to the study of the best arts solely by utility, and cultivate the liberal disciplines, let us see, I pray, whether mathematics are of no advantage in aiding the uses of human life, as the blind, base desire for profit of some has now broadcast with false preaching, so that those who embrace this faculty are openly mocked by the ignorant or by men occupied with other studies, as if they were losing their oil and labor on a useless and vain thing. Let us proceed therefore with, as they say, thick-witted Minerva, since we are dealing with those who can only be persuaded by the reason of profit, and let us brand this note onto a noble and generous discipline, so that by promising profit and riches, it may acquire for itself the studies and favor of such men. Let them first deny, if they can, that the mathematical arts have any popular utility, if trade—with the exercise of which so many are kept busy for the great occasion of profit—can be handled without arithmetic. Let them try then if they can measure anything without the aid of Geodesy. Let them plow the seas and seek distant regions, let them seek a new world, unsupported by the aid of nautical Astrology. What of the physician? How much does even one owe to Astronomy by Hippocrates' judgment, by whose guidance he knows the courses of the stars and especially of the moon? From this depends the entire ratio of the days which they call critical, against which one must diligently take care not to vex the sick patient with some more severe treatment while the moon, and that especially at the beginning of the disease, proceeds from combustion, as they now say, to the degree of opposition. How much advantage and utility, finally, does Geometry, Arithmetic, and all the rest bring to public and private uses? Since no art, even of the lowliest, can achieve its end without the help of mathematics. This is easily evident to anyone who examines each one more accurately; and it would be proven by us without difficulty, if we were avoiding a long dispute about a certain matter. By color, shadow, position, rarity, and density of media, and refraction, through which we see various ornaments and admirable figures of things daily? And are we deceived while watching with great pleasure? But we have strayed; for with them one must deal only by utility. Why, therefore, optics and painters having been omitted, let mere advantages be brought forward. How, then, can they deny that mathematics are wonderfully valid for the universal utility of cities, both by measuring the times of actions and by demonstrating the various revolutions of the universe? The military art, which is the right hand of politics, by what reason—wishing, when it is numerous, to show a very small multitude, it forms camps or battle lines to the shape of a circle; but when it desires to show its forces, it forms them to the shape of a quadrangle—if not with the help of Geometry alone? How does it either besiege and capture the cities of enemies, or defend its own, unless