Geometry

This Science of Magnitude, its properties, conditions, and appurtenances, has commonly been called geometry by all philosophers, now and from the beginning. But, truly, with a name too base and scant for a science of such dignity and amplitude. And perhaps that name, by common and secret consent of all wise men, has hitherto been allowed to remain, so that it might carry with it a perpetual memory of the first and most notable benefit shown to common people by that science: which was, when boundaries and meres of land and ground were lost and confounded (as in Egypt, yearly, with the overflowing of the Nile, the greatest and longest river in the world), or ground bequeathed was to be assigned, or ground sold was to be laid out, or (when disorder prevailed) commons were distributed into severalties. For where, upon these and such like occasions, some by ignorance, some by negligence, some by fraud, and some by violence, did wrongfully limit, measure, encroach upon, or claim (by pretense of just content and measure) those lands and grounds: great loss, disquiet, murder, and war did full often ensue: till, by God's mercy and men's industry, the perfect Science of Lines, Planes, and Solids (like a divine justiciar) gave to every man his own. The people then, pleased by this art and greatly relieved in the just measuring of their lands, and other philosophers writing rules for land measuring, between them both thus confirmed the name Geometria, that is (according to the very etymology of the word) land measuring. In it, the people knew no further of the use of magnitudes but in planes; and the philosophers had no fit hearers or scholars among them, to disclose further to than of flat, plain geometry. And though these philosophers knew of further use, and best understood the etymology of the word, yet this name Geometria was by them applied generally to all sorts of magnitudes: except at other times by Plato and Pythagoras, when they would precisely declare their own doctrine. Then it was

Plato, Republic 7.

Geometria, with them, the study concerned with the plane. But well you may perceive by Euclid's Elements that our Science is more ample than to measure planes, and nothing less is taught in it (on purpose) than how to measure land. Another name, therefore, must be had for our mathematical Science of Magnitudes: which regards neither clod nor turf, neither hill nor dale, neither earth nor heaven, but is absolute Megethologia: not creeping on ground and dazzling the eye with pole, perch, rod, or line, but "lifting the heart above the heavens by invisible lines, and

(☞)

immortal beams meeting with the reflections of the incomprehensible light, and so procuring joy and unspeakable perfection." Of which true use of our Megethica, or Megethologia, Divine Plato seemed to have good taste and judgment, and (by the name of geometry) so noted it, and warned his scholars of it, as may clearly be seen in his seventh Dialogue of the Commonwealth. Where it is thus, rightly translated: Truly, says Plato, whoever has (even very little) tasted of geometry will not deny us this: that this Science is of another condition, quite contrary to that which they who are exercised in it speak of it. And there it follows, of our geometry: It must be confessed, says Plato, that [geometry] is learned for the knowing of that which is ever, and not of that which in time both is bred and is brought to an end. etc. Geometry is the knowledge of that which is everlasting. It will lift up, therefore (O Gentle Sir), our mind to the Truth, and by that means it will prepare the Thought to the philosophical love of wisdom: that we may turn or convert toward heavenly things [both mind and thought] which now, otherwise than becomes us, we cast down on base or inferior things. etc. Chiefly, therefore, commandment must be given that such as inhabit this most honorable City by no means despise geometry. For even those things [done by it] which in manner seem to be beside the purpose of geometry are of no small importance. etc. And besides the manifold uses of geometry in matters pertaining to war, he adds more, of second unpurposed fruit and commodity arising from geometry, saying: But also we know that for the more easy learning of all arts it much imports whether one has any knowledge in geometry or no. etc. Let us therefore make an ordinance or decree that this Science shall be learned by young men in the second place. This was Divine Plato's judgment, both of the purposed, chief, and perfect use of geometry, and of his second, dependent, derivative commodities. And for us, Christian men, a thousand thousand more occasions are, to have need of the help of

I. D. In this, I would gladly shake off the earthly name of geometry.

metaphysical contemplations, by which to train our imaginations and minds little by little to forsake and abandon the gross and corruptible objects of our outward senses, and to apprehend, by sure demonstrative doctrine, Mathematical Things. And by them, readily to be helped and conducted to conceive, discourse, and conclude of things Intellectual, Spiritual, eternal, and such as concern our everlasting Bliss: which otherwise (without special privilege of Illumination or Revelation from heaven) no mortal man's wit (naturally) is able to reach or to compass. And truly, by my small Talent (from above) I am able to prove and testify that the literal Text and order of our divine Law, Oracles, and Mysteries require more skill in Numbers and Magnitudes than (commonly) the expositors have uttered: but rather only (at most) so warned, and showed their own want tin this. (To name any is needless; and to note the places is here no place; but if I am duly asked, my answer is ready.) And without the literal, grammatical, mathematical, or natural truths of such places, perceived by good and certain Art, no spiritual sense (proper to those places by Absolute Theology) will depend thereon.

(☞)

"No man, therefore, can doubt but toward the attaining of knowledge incomparable, and Heavenly Wisdom: Mathematical Speculations, both of Numbers and Magnitudes: are means, aids, and guides: ready, certain, and necessary." From from now on, in this my Preface, will I frame my talk to Plato's fugitive Scholars; or rather to such as well can (and also will) use their outward senses, to the glory of God, the benefit of their Country, and their own secret contentment, or honest advancement, on this earthly Scaffold. To them I will orderly recite, describe, and declare a great Number of arts, from our two mathematical fountains, derived into the fields of Nature. By which such Seeds and Roots as lie deep hidden in the ground of Nature are refreshed, quickened, and provoked to grow, shoot up, flower, and give fruit, infinite and incredible. And these arts shall be such as depend upon the properties of Magnitudes more than upon Number. And by good reason we may call them arts, and derived mathematical arts: for (at this time) I define

An art.

An art to be a methodical complete doctrine, having abundance of sufficient and peculiar matter to deal with, by the allowance of the metaphysical philosopher: the knowledge of which is necessary to the human estate. And that I count,

Derived mathematical art.

A derived mathematical art, which by a mathematical demonstrative method, in Numbers or Magnitudes, orders and confirms its doctrine, as much and as perfectly as the subject matter will admit. And for that, I intend to use the name and property of a

A mechanician.

mechanician, otherwise than (hitherto) it has been used, I think it good (for distinction's sake) to give you also a brief description of what I mean by that. A mechanician, or a mechanical workman, is he whose skill is, without knowledge of mathematical demonstration, perfectly to work and finish any sensible work, by the mathematician, principal or derivative, demonstrated or demonstrable. Full well I know that he who invents or makes these demonstrations is generally called a speculative mechanician, which differs nothing from a mechanical mathematician. So, in respect of diverse actions, one man may have the names of sundry arts: as sometimes a logician, sometimes (in the same matter otherwise handled) a rhetorician. Of these trifles I make (as now, in respect of my Preface) small account: to file them for the fine handling of subtle curious disputers. In other places they may command me to give good reason: and yet here I will not be unreasonable.

(1.)

First, then, from the purity, absoluteness, and immateriality of Principal Geometry is derived that kind of geometry which vulgarly is counted geometry: and is the Art of measuring sensible magnitudes, their just quantities and contents.

This science of magnitude, its properties, conditions, and appurtenances, has commonly been called geometry, now and from the beginning, by all philosophers. But truly, that name is too base and too small for a science of such dignity and breadth. And perhaps that name has been allowed to remain, by the common and secret agreement of all wise people, so that it carries a perpetual memory of the first and greatest benefit that this science showed to common people: which was when the boundaries and markers of land were lost and confused (as in Egypt every year, with the overflowing of the Nile, the greatest and longest river in the world), or when bequeathed land had to be assigned, or sold land had to be laid out, or (when disorder prevailed) commons had to be divided into separate holdings. For, on these and similar occasions, some by ignorance, some by negligence, some by fraud, and some by violence, wrongly limited, measured, encroached on, or claimed (under the pretense of a fair content and measure) those lands and grounds: great loss, unrest, murder, and war very often followed. Until, by God's mercy and human effort, the perfect science of lines, planes, and solids (like a divine judge) gave to each person what was their own. The people, pleased by this art and greatly helped by the fair measuring of their lands, and other philosophers writing rules for land measuring, between them both confirmed the name Geometria: that is, according to the very etymology of the word, land measuring. In this, the people knew no further use of magnitudes than in planes; and the philosophers had no fit hearers or students to disclose anything further to them than flat, plain geometry. And though these philosophers knew of further use and best understood the etymology of the word, yet they applied the name Geometria generally to all sorts of magnitudes: unless, at times, Plato and Pythagoras, when they wanted to state their own teaching precisely. Then, was

(* Plato, Republic 7.)

* Geometria, with them, the study concerned with the plane. But you can see clearly enough from Euclid's Elements that our science is wider than measuring planes, and that nothing less is taught in it (on purpose) than how to measure land. So another name must be had for our mathematical science of magnitudes: one that cares nothing for clod or turf, hill or dale, earth or heaven, but is absolute Megethologia: not creeping along the ground and dazzling the eye with pole, perch, rod, or line, but "lifting the heart above the heavens by invisible lines, and

immortal beams meet with the reflections of the incomprehensible light, and so bring joy and unspeakable perfection." Of this true use of our Megethica, or Megethologia, divine Plato seemed to have good taste and judgment: and (by the name of geometry) he noted it in that way, and warned his students of it, as may clearly be seen in his seventh dialogue, of the Commonwealth. Where it is thus, and well translated: Truly (says Plato) whoever has tasted even a very little of geometry will not deny us this: that this science is of a quite contrary condition to what those who are occupied in it say about it. And it follows, of our geometry: It must be confessed (says Plato) that [geometry] is learned for the knowledge of that which always is, and not of that which, in time, both comes into being and comes to an end. etc. Geometry is the knowledge of what is everlasting. It will therefore lift up our mind to the Truth (O Gentle Sir), and by that means prepare the thought for the philosophical love of wisdom: so that we may turn or convert toward heavenly things [both mind and thought], which now, contrary to what is fitting, we cast down onto base and inferior things. etc. Commandment must therefore chiefly be given, that those who inhabit this most honorable City by no means despise geometry. For even those things [done by it] which seem, in a manner, to lie beside the purpose of geometry are of no small importance. etc. And besides the manifold uses of geometry in matters relating to war, he adds more: a second, unintended fruit and benefit arising from geometry, saying: But also we know that, for the easier learning of all arts, it much matters whether one has any knowledge of geometry or not. etc. Let us therefore make an ordinance or decree, that this science be learned by young men in the second place. This was divine Plato's judgment, both of the intended, chief, and perfect use of geometry, and of its second, dependent, derivative benefits. And for us, Christians, there are a thousand thousand more occasions to need the help of *

(I. D. * In this, I would gladly shake off the earthly name of geometry.)

Megethologicall Contemplations: by which to train our imaginations and minds, little by little, to forsake and abandon the gross and corruptible objects of our outward senses; and to grasp, by sure demonstrative teaching, mathematical things. And by them, to be readily helped and guided to conceive, reason about, and conclude on things intellectual, spiritual, eternal, and such as concern our everlasting bliss: which otherwise (without a special privilege of illumination or revelation from heaven) no mortal person's wit can naturally reach or grasp. And truly, by my small talent from above, I am able to prove and testify that the literal text and order of our divine Law, Oracles, and Mysteries require more skill in numbers and magnitudes than expositors have commonly stated: they have rather only, at most, warned of this and shown their own lack in it. (To name any is needless, and to note the places is no part of this: but if I am properly asked, my answer is ready.) And without the literal, grammatical, mathematical, or natural truths of such places, perceived by good and certain art, no spiritual sense (proper to those places, by absolute theology) will depend on them.

"No one, therefore, can doubt that, toward the attaining of incomparable knowledge and heavenly wisdom, mathematical speculations, both of numbers and magnitudes, are means, aids, and guides: ready, certain, and necessary." From now on, in this preface of mine, I will shape my talk for Plato's runaway students: or rather, for those who can well, and also will, use their outward senses, to the glory of God, the benefit of their country, and their own private contentment or honest advancement on this earthly stage. To them I will orderly recite, describe, and declare a great number of arts, derived from our two mathematical fountains into the fields of Nature. By which such seeds and roots as lie deep hidden in the ground of Nature are refreshed, quickened, and provoked to grow, shoot up, flower, and bear fruit, infinite and incredible. And these arts will be such as depend more on the properties of magnitudes than on number. And with good reason we may call them arts, and mathematically derivative arts: for (at this time) I define

(An Art.)

An art, to be a methodical and complete body of teaching, having plenty of sufficient and particular matter to deal with, by the allowance of the metaphysical philosopher: knowledge of which is necessary to human life. And that I count,

(A Mathematically Derivative Art.)

An art mathematically derivative, which by a mathematical demonstrative method, in numbers or magnitudes, orders and confirms its teaching, as much and as perfectly as the subject matter allows. And because I intend to use the name and property of a

(A Mechanician.)

Mechanician, otherwise than it has been used so far, I think it good (for the sake of distinction) to give you also a brief description of what I mean by it. A mechanician, or a mechanical workman, is a person whose skill is to work and finish any sensible work perfectly, without knowledge of mathematical demonstration, by the principal or derivative mathematician, demonstrated or demonstrable. I know very well that whoever invents or makes these demonstrations is generally called a speculative mechanician, which differs nothing from a mechanical mathematician. So, in view of different actions, one person may hold the names of several arts: at one time a logician, at another (when the same matter is handled otherwise) a rhetorician. Of these trifles I make little account (for now, in view of my preface): I leave them for the fine handling of subtle, exacting disputers. In other places they may command me to give a good reason; but here I will not be unreasonable.

(1.)

First, then, from the purity, absoluteness, and immateriality of principal geometry is derived that kind of geometry which is commonly counted as geometry: and it is the art of measuring perceptible magnitudes, their exact quantities and contents.