Chapter X (1)

OF TRANSPARENCE, REFRACTION, AND OF THE POWER
OF THE EARTH TO PRODUCE LIVING CREATURES.

_A._ Thinking upon what you said yesterday, it looked like a generation of living creatures. I saw the love between the loadstone and the iron in their mutual attraction, their engendering in their close and contrary motion, and their issue in the iron, which being touched, hath the same attractive virtue. Now seeing they have the same internal motion of parts with that of the earth, why should not their substance be the same, or very near a-kin?

_B._ The most of them, if not all, that have written on this subject, when they call the loadstone a terrella, seem to think as you do. But I, except I could find proof for it, will not affirm it. For the earth attracteth all kind of bodies but air, and the loadstone none but iron. The earth is a star, and it were too bold to pronounce any sentence of its substance, especially of the planets, that are so lapt up in their several coats, as that they cannot work on our eyes, or any organ of our other senses.

_A._ I come therefore now to the business of the day. Seeing all generation, augmentation, and alteration is local motion, how can a body not transparent be made transparent?

_B._ I think it can never be done by the art of man. For as I said of hard and heavy bodies in the creation, so I think of diaphanous, that the very same individual body which was not transparent then, shall never be made transparent by human art.

_A._ Do not you see that every day men make glass, and other diaphanous bodies not much inferior in beauty to the fairest gems?

_B._ It is one thing to make one transparent of many by mixture, and another to make transparent of not transparent. Any very hard stone, if it be beaten into small sands, such as is used for hour-glasses, every one of those sands, if you look upon it with a microscope, you will find to be transparent; and the harder and whiter a stone is, so much the more transparent, as I have seen in the stone of which are made millstones, which stone is here called greet. And I doubt not but the sands of white marble must be more transparent. But there are no sands so transparent that they have not a scurf upon them, as hard, perhaps, as the stone itself; which they whose profession it is to make glass, have the art to scour and wash away. And therefore I think it no great wonder to bring those sands into one lump, though I know not how they do it.

_A._ I know they do it with lye made with a salt extracted from the ashes of an herb, of which salt they make a strong lye, and mingle it with the sand, and then bake it.

_B._ Like enough. But still it is a compound of two transparent bodies, whereof one is the natural stone, the other is the mortar. This therefore doth not prove, that one and the same body of not transparent can be made transparent.

_A._ Since they can make one transparent body of many, why do they not of a great many small sparks of natural diamond compound one great one? It would bear the charges of all the materials, and beside, enrich them.

_B._ It is probable it would. But it may be they know no salt that howsoever prepared, which, with how great a fire soever, can make them melt. And, it may be, the true crystal of the mountain, which is found in great pieces in the Alps, is but a compound of many small ones, and made by the earth’s annual motion; for it is a very swift motion. Suppose now that within a very small cavern of those rocks whose smallest atoms are crystal, and the cavity filled with air; and consider what a tumult would be made by the swift reciprocation of that air; whether it would not in time separate those atoms from the rock, and jumbling them together make them rub off their scurf from one another, and by little and little to touch one another in polished planes, and consequently stick together, till in length of time they become one lump of clean crystal.

_A._ I believe that the least parts of created substances lay mingled together at first, till it pleased God to separate all dissimilar natures, and congregate the similar, to which this annual motion is proper. But they say that crystal is found in the open air hanging like icicles upon the rocks, which, if true, defeats this supposition of a narrow cavern, and therefore I must have some farther experience of it before I make it my opinion. But howsoever, it still holds true that diaphanous bodies of all sorts, in their least parts, were made by God in the beginning of the world. But it may be true, notwithstanding those icicles. For the force of the air that could break off those diaphanous atoms in a cavern, can do the same in the open air. And I know that a less force of air can break some bodies into small pieces, not much less hard than crystal, by corrupting them.

_B._ That which you now have said is somewhat. But I deny not the possibility, but only doubt of the operation. You may therefore pass to some other question.

_A._ Well, I will ask you then a question about refraction. I know already that for the cause of refraction, when the light falleth through a thinner medium upon a thicker, you assign the resistance of the thicker body; but you do not mean there, by _rarum_ and _densum_, two bodies whereof in equal spaces one has more substance in it than the other.

_B._ No; for equal spaces contain equal bodies. But I mean by _densum_ any body which more resisteth the motion of the air, and by _rarum_ that which resisteth less.

_A._ But you have not declared in what that resistance consisteth.

_B._ I suppose it proceedeth from the hardness.

_A._ But from thence it will follow, that all transparent bodies that equally refract are equally hard, which I think is not true, because the refraction of glass is not greater, at least in comparison of their hardnesses, than that of water.

_B._ I confess it. Therefore I think we must take in gravity to a share in the production of this refraction. For I never considered refraction but in glass, because my business then was only to find the causes of the phenomena of telescopes and microscopes. Let therefore A B (in fig. 7) be a hard, and consequently, a heavy body; and from above, as from the sun, let C A be the line of incidence, and produced to D; and draw A E perpendicular to A B. It is manifest that the hardness in A B shall turn the stream of the light inwards toward A E, suppose in the line A _e_. It is also evident that the endeavour in B, which is, being heavy, downward, shall turn the stream again inward, towards A E, as in A _b_. Thus it is in refraction from the sun downwards. In like manner, if the light come from below, as from a candle in the point D, the line of incidence will be D A, and produced will pass to C. And the resistance of the hardness in A will turn the stream A C inward, suppose into A _l_, and make C _l_ equal to D _e_. For passing into a thinner medium, it will depart from the perpendicular in an angle equal to the angle D A _e_, by which it came nearer to it in A _e_. So also the resistance of the gravity in the point A shall turn the stream of the light into the line A _i_, and make the angle _l_ A _i_ equal to the angle _e_ A _b_. And thus you see in what manner, though not in what proportion, hardness and gravity conjoin their resistance in the causing of refraction.

_A._ But you proved yesterday, that a heavy body does not gravitate upon a body equally heavy. Now this A B has upper parts and lower parts; and if the upper parts do not gravitate upon the lower parts, how can there be any endeavour at all downward to contribute to the refraction?

_B._ I told you yesterday, that when a heavy body was set upon another body heavier or harder than itself, the endeavour of it downward was diverted another way, but not that it was extinguished. But in this case, where it lieth upon air, the first endeavour of the lowest part worketh downward. For neither motion nor body can be utterly extinguished by a less than an omnipotent power. All bodies, as long as they are bodies, are in motion one way or other, though the farther it be communicated, so much the less.

_A._ But since you hold that motion is propagated through all bodies, how hard or heavy soever they be, I see no cause but that all bodies should be transparent.

_B._ There are divers causes that take away transparency. First, if the body be not perfectly homogeneous, that is to say, if the smallest parts of it be not all precisely of the same nature, or do not so touch one another as to leave no vacuum within it; or though they touch, if they be not as hard in the contact as in any other line. For then the refractions will be so changed both in their direction, and in their strength, as that no light shall come through it to the eye; as in wood and ordinary stone and metal. Secondly, the gravity and hardness may be so great, as to make the angle refracted so great, as the second refraction shall not direct the beam of light to the eye; as if the angle of refraction were D A E, the refracted line would be perpendicular to A B, and never come to the line A D, in which is the eye.

_A._ To know how much of the refraction is due to the hardness, and how much to the gravity, I believe it is impossible, though the quantity of the whole be easily measured in a diaphanous body given. And both you and Mr. Warner have demonstrated, that as the sine of the angle refracted in one inclination is to the sine of the angle refracted in another inclination, so is the sine of one inclination to the sine of the angle of the other inclination. Which demonstrations are both published by Mersennus in the end of the first volume of his _Cogitata Physico-Mathematica_. But since there be many bodies, through which though there pass light enough, yet no object appear through them to the eye, what is the reason of that?

_B._ You mean paper. For paper windows will enlighten a room, and yet not show the image of an object without the room. But it is because there are in paper abundance of pores, through which the air passing moveth the air within; by the reflections whereof anything within may be seen. And in the same paper there are again as many parts not transparent, through which the air cannot pass, but must be reflected first to all parts of the object, and from them again to the paper; and at the paper either reflected again or transmitted, according as it falls upon pores or not pores; so that the light from the object can never come together at the eye.

_A._ There belongs yet to this subject the causes of the diversity of colours. But I am so well satisfied with that which you have written of it in the twenty-fourth chapter of your book _de Corpore_, that I need not trouble you farther in it. And now I have but one question more to ask you, which I thought upon last night. I have read in an ancient historian, that living creatures after a great deluge were produced by the earth, which being then very soft, there were bred in it, it may be by the rapid motion of the sun, many blisters, which in time breaking, brought forth, like so many eggs, all manner of living creatures great and small, which since it is grown hard it cannot do. What think you of it?

_B._ It is true that the earth produced the first living creatures of all sorts but man. For God said (Gen. i. 24), _Let the earth produce every living creature, cattle, and creeping thing, &c._ But then again (ver. 25) it is said that _God made the beast of the earth, &c._ So that it is evident that God gave unto the earth that virtue. Which virtue must needs consist in motion, because all generation is motion. But man, though the same day, was made afterward.

_A._ Why hath not the earth the same virtue now? Is not the sun the same as it was? Or is there no earth now soft enough?

_B._ Yes. And it may be the earth may yet produce some very small living creatures: and perhaps male and female. For the smallest creatures which we take notice of, do engender, though they do not all by conjunction; therefore if the earth produce living creatures at this day, God did not absolutely rest from all his works on the seventh day, but (as it is chap. ii. 2) _he rested from all the work he had made_. And therefore it is no harm to think that God worketh still, and when and where and what he pleaseth. Beside, it is very hard to believe, that to produce male and female, and all that belongs thereto, as also the several and curious organs of sense and memory, could be the work of anything that had not understanding. From whence, I think we may conclude, that whatsoever was made after the creation, was a new creature made by God no otherwise than the first creatures were, excepting only man.

_A._ They are then in an error that think there are no more different kinds of animals in the world now, than there were in the ark of Noah.

_B._ Yes, doubtless. For they have no text of Scripture from which it can be proved.

_A._ The questions of nature which I could yet propound are innumerable. And since I cannot go through them, I must give over somewhere, and why not here? For I have troubled you enough, though I hope you will forgive me.

_B._ So God forgive us both as we do one another. But forget not to take with you the demonstration of a straight line equal to an arc of a circle.

THE PROPORTION OF A STRAIGHT LINE TO HALF THE ARC OF A QUADRANT.

Describe the square A B C D, and divide it by the diagonals A C and B D, as also by the straight lines E G, F H, meeting in the centre I at right angles, into four equal parts. Then with the radius A B describe the quadrant B D cutting E G in K, and the diagonal A C in L; and so B L will be half the arc B D, equal to which we are to find a straight line. Divide I C into halves at M, and draw B M cutting E G in _a_. I say B M is equal to the arc B L. For the demonstration whereof we are to assume certain known truths and dictates of common-sense.

1. That the arc B K is the third part of the arc B D, and consequently two-thirds of the arc B L, and B K to K L as two to one.

2. That if a straight line be equal to the arc B L, and one end in B, the other will be somewhere in I C, and higher than the point L.

3. That wheresoever it be, two-thirds of it must be equal to the arc B K, and one-fifth to the arc K L.

4. That the arc of a quadrant described in the third part of the radius, or of E G, is equal to the third part of the arc B D, viz. to the arc B K. I may therefore call a third part of E G, the radius of B K; and a sixth part of E G, the radius of the arc K L, &c.

5. And lastly, that any straight line drawn from B to I C, if it be equal to the arc B L, it must cut the half radius I G, whose quadrantal arc is B L, into the proportion of two to one. For as the whole arc to the whole E G, so are the parts of it to the parts of E G.

These premises granted, which I think cannot be denied, I say again, that the straight line B M is equal to the arc B L.

DEMONSTRATION.

Because B I is to I M, by construction, as two to one, and the line I G divides the angle B I C in the midst, B _a_ will be to _a_ M as two to one, that is to say, as the arc B K to the arc K L. From the point M to the side B C erect a perpendicular M N. And because C M is half C I, the line M N will be half G C; and B N will be three-quarters of B C; and the square of B M equal to ten squares of a quarter of B C; and because B M is to B _a_ as three to two, M N will be to _a_ G as three to two. But M N is a quarter of E G, therefore _a_ G is two-thirds of a quarter of E G; that is, one-third of I G; that is, one-sixth of the whole E G. And I _a_ one-third of E G. Therefore I _a_ is the radius of the arc B K; and _a_ G the radius of the arc K L; and E G the radius of the whole arc B L D. Lastly, if a straight line be drawn from B to any other point of the line I C, though any line may be divided into the proportion of two to one, it shall not pass through the point _a_, and therefore not divide the radius of B L, which is I G, into the proportion of two to one. Therefore no straight line can be drawn from B to I C, except B M, so as to be equal to the arc B L. Therefore the straight line B M and the arc B L are equal.

Hence it follows, that seeing the square of B M is equal to ten squares of a quarter of B C, that a straight line equal to the quadrantal arc B L D is equal to ten squares of half the radius, as I have divers ways demonstrated heretofore.

SIX LESSONS
TO THE
PROFESSORS OF THE MATHEMATICS,

ONE OF GEOMETRY, THE OTHER OF ASTRONOMY,
IN THE CHAIRS SET UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE, IN THE
UNIVERSITY OF OXFORD.

TO THE RIGHT HONOURABLE

HENRY LORD PIERREPONT,

VISCOUNT NEWARK, EARL OF KINGSTON, AND
MARQUIS OF DORCHESTER.

MY MOST NOBLE LORD,

Not knowing on my own part any cause of the favour your Lordship has been pleased to express towards me, unless it be the principles, method, and manners you have observed and approved in my writings; and seeing these have all been very much reprehended by men, to whom the name of public professors hath procured reputation in the university of Oxford, I thought it would be a forfeiture of your Lordship’s good opinion, not to justify myself in public also against them, which, whether I have sufficiently performed or not in the six following Lessons addressed to the same professors, I humbly pray your Lordship to consider. The volume itself is too small to be offered to you as a present, but to be brought before you as a controversy it is perhaps the better for being short. Of arts, some are demonstrable, others indemonstrable; and demonstrable are those the construction of the subject whereof is in the power of the artist himself, who, in his demonstration, does no more but deduce the consequences of his own operation. The reason whereof is this, that the science of every subject is derived from a precognition of the causes, generation, and construction of the same; and consequently where the causes are known, there is place for demonstration, but not where the causes are to seek for. Geometry therefore is demonstrable, for the lines and figures from which we reason are drawn and described by ourselves; and civil philosophy is demonstrable, because we make the commonwealth ourselves. But because of natural bodies we know not the construction, but seek it from the effects, there lies no demonstration of what the causes be we seek for, but only of what they may be.

And where there is place for demonstration, if the first principles, that is to say, the definitions contain not the generation of the subject, there can be nothing demonstrated as it ought to be. And this in the three first definitions of Euclid sufficiently appeareth. For seeing he maketh not, nor could make any use of them in his demonstrations, they ought not to be numbered among the principles of geometry. And Sextus Empiricus maketh use of them (misunderstood, yet so understood as the said professors understand them) to the overthrow of that so much renowned evidence of geometry. In that part therefore of my book where I treat of geometry, I thought it necessary in my definitions to express those motions by which lines, superficies, solids, and figures, were drawn and described, little expecting that any professor of geometry should find fault therewith, but on the contrary supposing I might thereby not only avoid the cavils of the sceptics, but also demonstrate divers propositions which on other principles are indemonstrable. And truly, if you shall find those my principles of motion made good, you shall find also that I have added something to that which was formerly extant in geometry.

For first, from the seventh chapter of my book _De Corpore_, to the thirteenth, I have rectified and explained the principles of the science; _id est_, I have done that business for which Dr. Wallis receives the wages. In the seventh, I have exhibited and demonstrated the proportion of the parabola and parabolasters to the parallelograms of the same height and base; which, though some of the propositions were extant without that demonstration, were never before demonstrated, nor are by any other than this method demonstrable.

In the eighteenth, as it is now in English, I have demonstrated, for anything I yet perceive, equation between the crooked line of a parabola or any parabolaster and a straight line.

In the twenty-third I have exhibited the centre of gravity of any sector of a sphere.

Lastly, the twenty-fourth, which is of the nature of refraction and reflection, is almost all new.

But your Lordship will ask me what I have done in the twentieth, about the quadrature of the circle. Truly, my Lord, not much more than before. I have let stand there that which I did before condemn, not that I think it exact, but partly because the division of angles may be more exactly performed by it than by any organical way whatsoever; and I have attempted the same by another method, which seemeth to me very natural, but of calculation difficult and slippery. I call them only aggressions, retaining nevertheless the formal manner of assertion used in demonstration. For I dare not use such a doubtful word as _videtur_, because the professors are presently ready to oppose me with a _videtur quod non_. Nor am I willing to leave those aggressions out, but rather to try if it may be made pass for lawful, (in spite of them that seek honour, not from their own performances, but from other men’s failings), amongst many difficult undertakings carried through at once to leave one and the greatest for a time behind; and partly because the method is such as may hereafter give farther light to the finding out of the exact truth.

But the principles of the professors that reprehend these of mine, are some of them so void of sense, that a man at the first hearing, whether geometrician or not geometrician, must abhor them. As for example:

1. That two equal proportions are not double to one of the same proportions.

2. That a proportion is double, triple, &c. of a number, but not of a proportion.

3. That the same body, without adding to it, or taking from it, is sometimes greater, and sometimes less.

4. That a quantity may grow less and less eternally, so as at last to be equal to another quantity; or, which is all one, that there is a last in eternity.

5. That the nature of an angle consisteth in that which lies between the lines that comprehend the angle in the very point of their concourse, that is to say, an angle is the superficies which lies between the two points which touch, or, as they understand a point, the superficies that lies between the two nothings which touch.

6. That the quotient is the proportion of the division to the dividend.

Upon these and some such other principles is grounded all that Dr. Wallis has said, not only in his _Elenchus_ of my geometry, but also in his treatises of the _Angle of Contact_, and in his _Arithmetica Infinitorum_; which two last I have here in two or three leaves wholly and clearly confuted. And I verily believe that since the beginning of the world, there has not been, nor ever shall be, so much absurdity written in geometry, as is to be found in those books of his; with which there is so much presumption joined, that an ἀποκατάϛασις of the like conjunction cannot be expected in less than a Platonic year. The cause whereof I imagine to be this, that he mistook the study of _symbols_ for the study of _geometry_, and thought symbolical writing to be a new kind of method, and other men’s demonstrations set down in symbols new demonstrations. The way of analysis by squares, cubes, &c., is very ancient, and useful for the finding out whatsoever is contained in the nature and generation of rectangled planes, which also may be found without it, and was at the highest in Vieta; but I never saw anything added thereby to the science of geometry, as being a way wherein men go round from the equality of rectangled planes to the equality of proportion, and thence again to the equality of rectangled planes, wherein the symbols serve only to make men go faster about, as greater wind to a windmill.

It is in sciences as in plants; growth and branching is but the generation of the root continued; nor is the invention of theorems anything else but the knowledge of the construction of the subject prosecuted. The unsoundness of the branches are no prejudice to the roots, nor the faults of theorems to the principles. And active principles will correct false theorems if the reasoning be good; but no logic in the world is good enough to draw evidence out of false or unactive principles. But I detain your Lordship too long. For all this will be much more manifest in the following discourses, wherein I have not only explained and rectified many of the most important principles of geometry, but also by the examples of those errors which have been committed by my reprehenders, made manifest the evil consequence of the principles they now proceed on. So that it is not only my own defence that I here bring before you, but also a positive doctrine concerning the true grounds, or rather atoms of geometry, which I dare only say are very singular, but whether they be very good or not, I submit to your Lordship’s judgment. And seeing you have been pleased to bestow so much time, with great success, in the reading of what has been written by other men in all kinds of learning, I humbly pray your Lordship to bestow also a little time upon the reading of these few and short lessons; and if your Lordship find them agreeable to your reason and judgment, let me, notwithstanding the clamour of my adversaries, be continued in your good opinion, and still retain the honour of being,

My most noble Lord,
Your Lordship’s most
humble and obliged servant,
THOMAS HOBBES.

LONDON, _June 10, 1656_.

LESSONS

OF

THE PRINCIPLES OF GEOMETRY, &c.

TO THE EGREGIOUS PROFESSORS OF THE MATHEMATICS, ONE OF
GEOMETRY, THE OTHER OF ASTRONOMY, IN THE CHAIRS SET
UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE,
IN THE UNIVERSITY OF OXFORD.

LESSON I.

I suppose, most egregious professors, you know already that by geometry, though the word import no more but the measuring of land, is understood no less the measuring of all other quantity than that of bodies. And though the definition of geometry serve not for proof, nor enter into any geometrical demonstration, yet for understanding of the principles of the science, and for a rule to judge by, who is a geometrician, and who is not, I hold it necessary to begin therewith.

Geometry is the science of determining the quantity of anything, not measured, by comparing it with some other quantity or quantities measured. Which science therefore whosoever shall go about to teach, must first be able to tell his disciple what measuring or dimension is; by what each several kind of quantity is measured; what quantity is, and what are the several kinds thereof. Therefore as they, who handle any one part of geometry, determine by definition the signification of every word which they make the subject or predicate of any theorem they undertake to demonstrate; so must he which intendeth to write a whole body of geometry, define and determine the meaning of whatsoever word belongeth to the whole science. The design of Euclid was to demonstrate the properties of the five regular bodies mentioned by Plato; in which demonstrations there was no need to allege for argument the definition of quantity, which it may be was the cause he hath not anywhere defined it, but done what he undertook without it. And though having perpetually occasion to speak of measure, he hath not defined measure; yet instead thereof he hath, in the beginning of his first elements, assumed an axiom which serveth his turn sufficiently as to the measure of lines, which is the eighth axiom; that those things which lie upon one another all the way (called by him ἐφαρμόζοντα) are equal. Which axiom is nothing else but a description of the art of measuring length and superficies. For this ἐφάρμοσις can have no place in solid bodies, unless two bodies could at the same time be in one place. But amongst the principles of geometry universal, the definitions are necessary, both of quantity and dimensions.

Quantity is that which is signified by what we answer to him that asketh, _how much_ any thing is? and thereby determine the magnitude thereof. For magnitude being a word indefinite, if a man ask of a thing, _quantum est?_ that is, _how much_ it is, we do not satisfy him by saying it is magnitude or quantity, but by saying it is _tantum_, _so much_. And they that first called it in Greek, πηλικότης, and in Latin _quantity_, might more properly have called it in Latin _tantity_, and in Greek τηλικότης; and we, if we allowed ourselves the eloquence of the Greeks and Latins, should call it the _so-muchness_.

There is therefore to everything concerning which a man may ask without absurdity, _how much it is_, a certain quantity belonging, determining the magnitude to be _so much_. Also wheresoever there is _more_ and _less_, there is one kind of quantity or other. And first there is the quantity of bodies, and that of three kinds: length, which is by one way of measuring; superficies, made of the complication of two lengths, or the measure taken two ways; and solid, which is the complication of three lengths, or of the measure taken three ways, for breadth or thickness are but other lengths. And the science of geometry, so far forth as it contemplateth bodies only, is no more but by measuring the length of one or more lines, and by the position of others known in one and the same figure, to determine by ratiocination, how much is the superficies; and by measuring length, breadth, and thickness, to determine the quantity of the whole body. Of this kind of magnitudes and quantities the subject is body.

And because for the computing of the magnitudes of bodies, it is not necessary that the bodies themselves should be present, the ideas and memory of them supplying their presence, we reckon upon those imaginary bodies, which are the quantities themselves, and say the length is so great, the breadth so great, &c. which in truth is no better than to say the length is so long, or the breadth so broad, &c. But in the mind of an intelligent man it breedeth no mistake.

Besides the quantity of bodies, there is a quantity of time. For seeing men, without absurdity, do ask how much it is; by answering _tantum_, _so much_, they make manifest there is a quantity that belongeth unto time, namely, a length. And because length cannot be an accident of time, which is itself an accident, it is the accident of a body; and consequently the length of the time, is the length of the body; by which length or line, we determine how much the time is, supposing some body to be moved over it.

Also we not improperly ask with _how much_ swiftness a body is moved; and consequently there is a quantity of motion or swiftness, and the measure of that quantity is also a line. But then again, we must suppose another motion, which determineth the time of the former. Also of force, there is a question of _how much_, which is to be answered by _so much_; that is, by quantity. If the force consist in swiftness, the determination is the same with that of swiftness, namely, by a line; if in swiftness and quantity of body jointly, then by a line and a solid; or if in quantity of body only, as weight, by a solid only.

So also is number quantity; but in no other sense than as a line is quantity divided into equal parts.

Of an angle, which is of two lines whatsoever they be, meeting in one point, the digression or openness in other points, it may be asked how great is that digression? This question is answered also by quantity. An angle therefore hath quantity, though it be not the subject of quantity; for the body only can be the subject, in which body those straddling lines are marked.

And because two lines may be made to divaricate by two causes; one, when having one end common and immoveable, they depart one from another at the other ends circularly, and this is called simply an angle; and the quantity thereof is the quantity of the arch, which the two lines intercept.

The other cause is the bending of a straight line into a circular or other crooked line, till it touch the place of the same line, whilst it was straight, in one only point. And this is called an angle of contingence. And because the more it is bent, the more it digresseth from the tangent, it may be asked _how much_ more? And therefore the answer must be made by quantity; and consequently an angle of contingence hath its quantity as well as that which is called simply an angle. And in case the digression of two such crooked lines from the tangent be uniform, as in circles, the quantity of their digression may be determined. For, if a straight line be drawn from the point of contact, the digression of the lesser circle will be to the digression of the greater circle, as the part of the line drawn from the point of contact, and intercepted by the circumference of the greater circle is to the part of the same line intercepted by the circumference of the lesser circle, or, which is all one, as the greater radius is to the lesser radius. You may guess by this what will become of that part of your last book, where you handle the question of the quantity of an angle of contingence.

Also there lieth a question of _how much comparatively_ one magnitude is to another magnitude, as how much water is in a tun in respect of the ocean, how much in respect of a pint; _little_ in the first respect, _much_ in the latter. Therefore the answer must be made by some respective quantity. This respective quantity is called _ratio_ and proportion, and is determined by the quantity of their differences; and if their differences be compared also with the quantities themselves that differ, it is called simply proportion, or proportion geometrical. But if the differences be not so compared, then it is called proportion arithmetical. And where the difference is none, there is no quantity of the proportion, which in this case is but a bare comparison.

Also concerning heat, light, and divers other qualities, which have degrees, there lieth a question of _how much_, to be answered by a _so much_, and consequently they have their quantities, though the same with the quantity of swiftness: because the intensions and remissions of such qualities are but the intensions and remissions of the swiftness of that motion by which the agent produceth such a quality. And as quantity may be considered in all the operations of nature, so also doth geometry run quite through the whole body of natural philosophy.

To the principles of geometry the definition appertaineth also of a _measure_, which is this, _one quantity is the measure of another quantity, when it, or the multiple of it, is coincident in all points with the other quantity_. In which definition, instead of that ἐφαρμογὴ of Euclid, I put coincidence. For the superposition of quantities, by which they render the word ἐφαρμογὴ, cannot be understood of bodies, but only of lines and superficies. Nevertheless many bodies may be coincident successively with one and the same place, and that place will be their measure, as we see practised continually in the measuring of liquid bodies, which art of measuring may properly be called ἐφάρμοσις, but not superposition.

Also the definitions of _greater_, _less_, and _equal_, are necessary principles of geometry. For it cannot be imagined than any geometrician should demonstrate to us the equality and inequality of magnitudes, except he tell us first what those words do signify. And it is a wonder to me, that Euclid hath not anywhere defined what are equals, or at least, what are equal bodies, but serveth his turn throughout with that forementioned ἐφάρμοσις, which hath no place in solids, nor in time, nor in swiftness, nor in circular, or other crooked lines; and therefore no wonder to me, why geometry hath not proceeded to the calculation neither of crooked lines, nor sufficiently of motion, nor of many other things, that have proportion to one another.

Equal bodies, superficies, and lines, are those of which every one is capable of being coincident with the place of every one of the rest: and equal times, wherein with one and the same motion equal lines are described. And equally swift are those motions by which we run over equal spaces in any time determined by any other motion. And universally all quantities are equal, that are measured by the same number of the same measures.

It is necessary also to the science of geometry, to define what quantities are of one and the same kind, which they call _homogeneous_, the want of which definitions hath produced those wranglings (which your book _De Angulo Contactus_ will not make to cease) about the angle of contingence.

_Homogeneous_ quantities are those which may be compared by (ἐφάρμοσις) application of their measures to one another; so that solids and superficies are heterogeneous quantities, because there is no coincidence or application of those two dimensions.

No more is there of line and superficies, nor of line and solid, which are therefore heterogeneous. But lines and lines, superficies and superficies, solids and solids, are homogeneous.

Homogeneous also are line, and the quantity of time; because the quantity of time is measured by the application of a line to a line; for though time be no line, yet the quantity of time is a line, and the length of two times is compared by the length of two lines.

Weight and solid have their quantity homogeneous, because they measure one another by application, to the beam of a balance. Line and angle simply so called, have their quantity homogeneous, because their measure is an arch or arches of a circle applicable in every point to one another.

The quantity of an angle simply so called, and the quantity of an angle of contingence are heterogeneous. For the measures by which two angles simply so called are compared, are in two coincident arches of the same circle; but the measure by which an angle of contingence is measured, is a straight line intercepted between the point of contact and the circumference of the circle; and therefore one of them is not applicable to the other; and consequently of these two sorts of angles the quantities are heterogeneous. The quantities of two angles of contingence are homogeneous; for they may be measured by the ἐφάρμοσις of two lines, whereof one extreme is common, namely, the point of contact, the other extremes are in the arches of the two circles.

Besides this knowledge of what is quantity and measure, and their several sorts, it behoveth a geometrician to know why, and of what, they are called principles. For not every proposition that is evident is therefore a principle. A principle is the beginning of something. And because definitions are the beginnings or first propositions of demonstration, they are therefore called principles, principles, I say, of demonstration. But there be also necessary to the teaching of geometry other principles, which are not the beginnings of demonstration, but of construction, commonly called petitions; as that it may be granted _that a man can draw a straight line, and produce it; and with any radius, on any centre describe a circle_, and the like. For that a man may be able to describe a square, he must first be able to draw a straight line; and before he can describe an equilateral triangle, he must be able first to describe a circle. And these petitions are therefore properly called principles, not of demonstration, but of operation. As for the commonly received third sort of principles, called _common notions_, they are principles, only by permission of him that is the disciple; who being ingenuous, and coming not to cavil but to learn, is content to receive them, though demonstrable, without their demonstrations. And though definitions be the only principles of demonstration, yet it is not true that every definition is a principle. For a man may so precisely determine the signification of a word as not to be mistaken, yet may his definition be such as shall never serve for proof of any theorem, nor ever enter into any demonstration, such as are some of the definitions of Euclid, and consequently can be no beginnings of demonstration, that is to say, no principles.

All that hitherto hath been said, is so plain and easy to be understood, that you cannot, most egregious professors, without discovering your ignorance to all men of reason, though no geometricians, deny it. And the same (saving that the words are all to be found in dictionaries) new; also to him that means to learn, not only the practice, but also the science of geometry necessary, and, though it grieve you, mine. And now I come to the definitions of Euclid.

The first is of a point: Σημεῖον, &c. “_Signum est, cujus est pars nulla_,” that is to say, _a mark is that of which there is no part_. Which definition, not only to a candid, but also to a rigid construer, is sound and useful. But to one that neither will interpret candidly, nor can interpret accurately, is neither useful nor true. Theologers say the soul hath no part, and that an angel hath no part, yet do not think that soul or angel is a point. A mark or as some put instead of it ϛίγμη, which is a mark with a hot iron, is visible; if visible, then it hath quantity, and consequently may be divided into parts innumerable. That which is indivisible is no quantity; and if a point be not quantity, seeing it is neither substance nor quality, it is nothing. And if Euclid had meant it so in his definition, as you pretend he did, he might have defined it more briefly, but ridiculously, thus, _a point is nothing_. Sir Henry Savile was better pleased with the candid interpretation of Proclus, that would have it understood respectively to the matter of geometry. But what meaneth this _respectively to the matter of geometry_? It meaneth this, that no argument in any geometrical demonstration should be taken from the division, quantity, or any part of a point; which is as much as to say, a point is that whose quantity is not drawn into the demonstration of any geometrical conclusion; or, which is all one, whose quantity is not considered.

An accurate interpreter might make good the definition thus, _a point is that which is undivided_; and this is properly the same with _cujus non est pars_: for there is a great difference between _undivided_ and _indivisible_, that is, between _cujus non est pars_, and _cujus non potest esse pars_. Division is an act of the understanding; the understanding is therefore that which maketh parts, and there is no part where there is no consideration but of one. And consequently Euclid’s definition of a point is accurately true, and the same with mine, which is, that _a point is that body whose quantity is not considered_. And _considered_ is that, as I have defined it chap. I. at the end of the third article, which is not put to account in demonstration.

Euclid therefore seemeth not to be of your opinion, that say a point is nothing. But why then doth he never use this definition in the demonstration of any proposition? Whether he useth it expressly or no, I remember not; but the sixteenth proposition of the third book without the force of this definition is undemonstrated.

The second definition is of a line: γραμμὴ δὲ μῆκος ἂπλατες. “_Linea est longitudo latitudinis expers_; _a line is length which hath no breadth_;” and if candidly interpreted, sound enough, though rigorously not so. For to what purpose is it to say _length not broad_, when there is no such thing as a _broad length_. One path may be broader than another path, but not one mile than another mile; and it is not the path but the mile which is the way’s length. If therefore a man have any ingenuity he will understand it thus, _that a line is a body whose length is considered without its breadth_, else we must say absurdly a _broad length_; or untruly, that there be bodies which have length and yet no breadth; and this is the very sense which Apollonius, saith Proclus, makes of this definition; “when we measure,” says he, “the length of a way, we take not in the breadth or depth, but consider only one dimension.” See this of Proclus cited by Sir Henry Savile, where you shall find the very word _consider_.

The fourth definition is of a straight line, thus Ἐυθεῖα γραμμή ἐϛιν, &c. “_Recta linea est quæ ex æquo sua ipsius puncta inter jacet._” _A straight line is that which lieth equally (or perhaps evenly) between its own points._ This definition is inexcusable. Between what points of its own can a straight line lie but between its extremes? And how lies it evenly between them, unless it swerve no more from some other line which hath the same extremes, one way than another? And then why are not between the same points both the lines straight? How bitterly, and with what insipid jests would you have reviled Euclid for this, if living now he had written a _Leviathan_ . And yet there is somewhat in this definition to help a man, not only to conceive the nature of a straight line (for who doth not conceive it?) but also to express it. For he meant perhaps to call a straight line that which is all the way from one extreme to another, equally distant from any two or more such lines as being like and equal have the same extremes. So the axis of the earth is all the way equally distant from the circumference of any two or more meridians. But then before he had defined a straight line, he should have defined what lines are _like_, and what are _equal_. But it had been best of all, first to have defined crooked lines, by the possibility of a deduction or setting further asunder of their extremes; and then straight lines, by the impossibility of the same.

The seventh definition, which is that of a plain superficies, hath the same faults.

The eighth is of a plane angle, Ἑπὶπεδος γωνία ἑϛὶν ἡ ἐν ἐπιπέδω, &c. “_Angulus planus est duarum linearum in plano se mutuo tangentium, et non in directum jacentium, alterius ad alteram inclinatio._” _A plane angle is the inclination one towards another of two lines that touch one another in the same plane, and lie not in the same straight line._ Besides the faults here observed by Sir Henry Savile, as the clause of not lying in the same straight line, and the obscurity or equivocation of the word _inclination_, there is yet another, which is, that by this definition two right angles together taken, are no angle; which is a fault which you somewhere (asking leave to use the word _angle_, καταχριϛικώς acknowledge, but avoid not. For in geometry, where you confess there is required all possible accurateness, every καταχρῆσις is a fault. Besides you see by this definition, that Euclid is not of your, but of Clavius’s opinion. For it is manifest that the two lines which contain an angle of contact incline one towards another, and come together in a point, and lie not in the same straight line, and consequently make an angle.

The thirteenth definition is exact, but makes against your doctrine, that a point is nothing. Examine it. Ὅρος ἐϛὴν ὅ τινός ἐϛῖ πέρας. “_Terminus est quod alicujus extremum est._” _A term or bound is that which is the extreme of anything._ We had before, _the extremes of a line are points_. But what is in a line the extreme, but the first or last _part_, though you may make that part as small as you will? A point is therefore a part, and nothing is no extreme.

The fourteenth, Σχῆμα ἐϛὶ τὸ ὑπὸ τινος ἤ τινῶν ὅρων περιεχόμενον. “_Figura est (subaudi quantitas) quæ ab aliquo, vel aliquibus terminis undique continetur sive clauditur._” _A figure is quantity contained within some bound or bounds._ Or shortly thus, _a figure is quantity every way determined_, is in my opinion as exact a definition of a figure as can possibly be given, though it must not be so in yours. For this _determination_ is the same thing with _circumscription_; and whatsoever is anywhere _(ubicunque) definitivè_ is there also _circumscriptivè_; and by this means the distinction is lost, by which theologers, when they deny God to be in any place, save themselves from being accused of saying he is nowhere; for that which is nowhere is nothing. This definition of Euclid cannot therefore possibly be embraced by you that carry double, namely, mathematics and theology. For if you reject it, you will be cast out of all mathematic schools; and if you maintain it, from the society of all school-divines, and lose the thanks of the favour you have shown (you the astronomer) to Bishop Bramhall.