Pantometria — A Geometrical Treatize of Measuring

Although geometry, by its certainty, has such a privilege as few other sciences—being so fortified with demonstration that no precept or rule thereof can be reproved for lack of truth—yet, lest some, either of ignorance or malice, should affirm it unprofitable, not serving to any necessary use in a public commonwealth, I thought good to say somewhat thereof.

And first, the sentence of Plato written at the entry into his school comes to my remembrance: Latin: "Let no one ignorant of geometry enter here." original: "ἀγεωμέτρητος μηδεὶς εἰσίτω" thereby excluding all such as were ignorant of geometry as unable or unmeet to attain the higher secrets or mysteries of philosophy. Aristotle also, entreating of moral philosophy in the fifth book of his Ethics, most beautifully paints out justice with geometrical figures, discerning and severally comparing sundry parts thereof with geometrical and arithmetical proportions. But his interpreters, whose works are yet extant, for want of skill in these matters, have so blemished, darkened, and defaced his meaning that scarcely any resemblance thereof shall appear to the reader. In sundry other of his works also of natural philosophy, such as the Physics, Meteors, De Cœlo & Mundo Latin: "On the Heavens and the World.", etc., you shall find sundry demonstrations that, without geometry, cannot possibly be understood. And to leave philosophy, how necessary it is to attain exact knowledge in astronomy, music, perspective, cosmography, and navigation, with many other sciences and faculties, whoever travels but meanly therein shall soon find.

But to omit talking of geometry in general, and to speak privately of this treatise, I think it good first to open the order and effect thereof, and then somewhat to say of its applications. Like as every geometrical body is environed with surfaces, and every surface enclosed with lines—so that it consists of these three: longitude, latitude, and profundity depth, and without consideration of these three, no solid may be measured—so is this treatise also divided into three books. The first, entreating of lines, shows sundry means to measure all manner of lengths, heights, distances, and depths. The second, termed Planimetra, sets forth diverse means and rules to measure manifold surfaces—plain, convex, and concave—whether they be compounded with straight or circular lines, or mixed of both. In the third, named Stereometria, is set out the exact measurement of sundry solids, replenished with a number of rules and precepts gathered out of Euclid, Archimedes, and Apollonius Pergaeus his Conics, wherein the reader shall not a little delight himself with the fineness and subtlety of their inventions, especially if he endeavors himself to search out the reason, cause, and demonstration of them.

And now, somewhat to speak of the commodity of these conclusions: as the skillful in architecture can apply the Stereometria to serve his turn in pre-ordination and forecasting both the charges, quantities, and proportion of all parcels necessarily appertaining to any kind of buildings, so Planimetra may serve for disposing of all manner of ground, plots of cities, towns, forts, castles, palaces, or other edifices. The marshal of the field shall thereby speedily appoint a place convenient for his camp, distributing every part thereof according to the number of his men, horse, carriage, etc. Also in surveying, parting, and dividing of lands and woods, it is most requisite as well for exact as speedy dispatch therein. Of this, we have notable record in histories, how much this science availed the Egyptians when, by the inundation of the Nile, their whole country was so drowned that with the slime of the water all their bounds and marks were defaced; yet certain wise men, aided with knowledge in this science, found out and distributed to every man his own.

The other part, named Longimetra, the ingenious practitioner will apply to topography, fortification, conducting of mines under the earth, and shooting of great ordnance. So that, as there is no kind of man, of what vocation or degree soever he be, but shall find matter both to exercise his wit and diversely to pleasure himself, so surely for a gentleman especially that professes the wars—as well for discoveries made by sea, as fortification, placing of camps, and conducting of armies on the land—how necessary it is to be able exactly to describe the true plots...