CHAPTER X.. Of Transparence, Refraction, and of the Power

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

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.

The fifteenth is of a circle. Κοὐκλος ἐστὶ σχῆμα ἐπίπεδον, &c. A circle is a plain figure comprehended by one line which is called the circumference, to which circumference all the straight lines drawn from one of the points within the figure are equal to one another. This is true. But if a man had never seen the generation of a circle by the motion of a compass or other equivalent means, it would have been hard to persuade him that there was any such figure possible. It had been therefore not amiss first to have let him see that such a figure might be described. Therefore so much of geometry is no part of philosophy, which seeketh the proper passions of all things in the generation of the things themselves.

After the fifteenth till the last or thirty-fifth definition, all are most accurate, but the last which is this, parallel straight lines are those which being in the same plane, though infinitely produced both ways, shall never meet. Which is less accurate. For how shall a man know that there be straight lines which shall never meet, though both ways infinitely produced? Or how is the definition of parallels, that is, of lines perpetually equidistant, good, wherein the nature of equidistance is not signified? Or if it were signified, why should it not comprehend as well the parallelism of circular and other crooked lines, as of straight, and as well of superficies, as of lines? By parallels is meant equidistant both lines and superficies, and the word is therefore not well defined without defining first equality of distance. And because the distance between two lines or superficies, is the shortest line that can join them, there either ought to be in the definition the shortest distance, which is that of the perpendicular and without inclination, or the distance in equal inclination, that is, in equal angles. Therefore if parallels be defined to be those lines or superficies, where the lines drawn from one to another in equal angles be equal, the definition, as to like lines, or like superficies, will be universal and convertible. And if we add to this definition, that the equal angles be drawn not opposite ways, it will be absolute, and convertible in all lines and superficies; and the definition will be this: parallels are those lines and superficies between which every line drawn, in any angle, is equal to any other line drawn in the same angle the same way. For by this definition the distance between them will perpetually be equal, and consequently they will never come nearer together, how much, or which way soever they be produced. And the converse of it will be also true, if two lines, or two superficies be parallel, and a straight line be drawn from one to the other, any other straight line, drawn from one to the other in the same angle, and the same way, will be equal to it. This is manifestly true, and, most egregious professors, new, at least to you.

And thus much for the definitions placed before the first of Euclid’s Elements.

Before the third of his Elements is this definition: “In circulo æqualiter distare a centro rectæ lineæ dicuntur, cum perpendiculares quæ a centro in ipsas ducuntur sunt æquales.” In a circle two straight lines are said to be equally distant from the centre, upon which the perpendiculars drawn from the centre are equal. This is true; but it is rather an axiom than a definition, as being demonstrable that the perpendicular is the measure of the distance between a point and a straight or a crooked line.

Before the fifth Element the first definition is of a part: Pars est magnitudo magnitudinis, minor majoris, cum minor metitur majorem. A part is one magnitude of another, the less of the greater, when the less measureth the greater. From which definition it followeth, that more than a half is not a part of the whole. But because Euclid meaneth here an aliquot part, as a half, a third, or a fourth, &c., it may pass for the definition of a measure under the name of part, as thus: a measure is a part of the whole, when multiplied it may be equal to the whole, though properly a measure is external to the thing measured, and not the aliquot part itself, but equal to an aliquot part.

But the third definition is intolerable; it is the definition of λόγος, in Latin ratio, in English, proportion, in this manner, λόγος ἐςὶ δύο μεγεθῶν ὁμογενῶν ῆ κατὰ πηλικότητα προς ἄλληλα ποιὰ σχέσις. “Ratio est duarum magnitudinum ejusdem generis mutua quædam secundum quantitatem habitudo.” Proportion is a certain mutual habitude in quantity, of two magnitudes of the same kind, one to another. First, we have here ignotum per ignotius; for every man understandeth better what is meant by proportion than by habitude. But it was the phrase of the Greeks when they named like proportions, to say, the first to the second, οὕτως ἔχει, id est, ita se habet, and in English, is as, the third to the fourth. As for example, in the proportions of two to four, and three to six, to say two to four, οὕτως ἔχει, id est, ita se habet, id est, is as, three to six. From which phrase Euclid made this his definition of proportion by ποιὰ σχέσις, which the Latins translate quædam habitudo. Quædam in a definition is a most certain note of not understanding the word defined; and in Greek, ποιὰ σχέσις is much worse; for to render rightly the Greek definition, we are to say in English, that proportion is a what-shall-I-call-it-isness, or soness of two magnitudes, &c.; than which nothing can be more unworthy of Euclid. It is as bad as anything was ever said in geometry by Orontius, or by Dr. Wallis. That proportion is quantity compared, that is to say, little or great in respect of some other quantity, as I have above defined it, is I think intelligible.

The fourth is, Ἀναλογία δέ ὲστιν ῆ των λόγων ὁμοιότης. “Proportio vero est rationum similitudo.” Here we have no one word by which to render Ἀναλογία; for our word proportion is already bestowed upon the rendering of λόγος. Nevertheless the Greek may be translated into English thus, iterated proportions. But iterated proportion is the same with eadem ratio. To what purpose then serveth the sixth definition, which is of eadem ratio? For Ἀναλογία and eadem ratio and similitudo rationum, are the same thing, as appeareth by Euclid himself, where he defines those quantities, that are in the same proportion by ἀνάλογον. Therefore the sixth definition is but a lemma, and assumed without demonstration.

The fourteenth, “Compositio rationis est sumptio antecedentis cum consequente, ceu unius, ad ipsum consequentem,” To compound proportion, is to take both antecedent and consequent together as one magnitude, and compare it to the consequent, is good; though he might have compared it as well with the antecedent; for both ways it had been a composition of proportion. We are to note here, that the composition defined in this place by Euclid is not adding together of proportions, but of two quantities that have proportion. And therefore it is not the same composition which he defineth in the fourth place before the sixth element, for there he defineth the addition of one proportion to another proportion in this manner: λόγος ἐκ λόγων συγκεῖσθαι λέγεται, &c. A proportion is said to be compounded of proportions, when their quantities multiplied into one another make a proportion; as when we would compound or add together the proportions of three to two, and of four to five, we must multiply three and four, which maketh twelve, and two and five, which maketh ten. And then the proportion of twelve to ten is the sum of the proportions of three to two, and of four to five, which is true, but not a definition; for it may and ought to be demonstrated. For to define what is addition of two proportions (which are always in four quantities, though sometimes one of them be twice named) we are to say, that they are then added together when we make the second to another in the same proportion, which the third hath to the fourth.

And thus much of the definitions; of which some, very few, you see are faulty; the rest either accurate, or good enough if well interpreted. For the rest of the elements all are accurate, notwithstanding that you allow not for good any definition in geometry that hath in it the word motion, of which there be divers before the eleventh Element. But I must here put you in mind, that geometry being a science, and all science proceeding from a precognition of causes, the definition of a sphere, and also of a circle, by the generation of it, that is to say, by motion, is better than by the equality of distance from a point within.

The second sort of principles are those of construction, usually called postulata, or petitions. As for those notiones communes, called axioms, they are from the definitions of their terms demonstrable, though they be so evident as they need not demonstration. These petitions are by Euclid called Ἀιτήματα, such as are granted by favour, that is, simply petitions, whereas by axiom is understood that which is claimed as due. So that between Ἀξίωμα and Ἀίτημα there is this other difference, that this latter is simply a petition, the former a petition of right.

Of petitions simply, the first is, that from any point to any point may be drawn a straight line. The second, that a finite straight line may be produced. The third, that upon any centre at any distance may be described a circle. All which are both evident and necessary to be granted.

And by all these a man may easily perceive that Euclid in the definitions of a point, a line, and a superficies, did not intend that a point should be nothing, or a line be without latitude, or a superficies without thickness; for if he did, his petitions are not only unreasonable to be granted, but also impossible to be performed. For lines are not drawn but by motion, and motion is of body only. And therefore his meaning was, that the quantity of a point, the breadth of a line, and the thickness of a superficies were not to be considered, that is to say, not to be reckoned in the demonstration of any theorems concerning the quantity of bodies, either in length, superficies, or solid.

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OF THE FAULTS THAT OCCUR IN DEMONSTRATION.

TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN THE UNIVERSITY OF OXFORD.

LESSON II.

There be but two causes from which can spring an error in the demonstration of any conclusion in any science whatsoever; and those are ignorance or want of understanding, and negligence. For as in the adding together of many and great numbers, he cannot fail that knoweth the rules of addition, and is also all the way so careful, as not to mistake one number or one place for another; so in any other science, he that is perfect in the rules of logic, and is so watchful over his pen, as not to put one word for another, can never fail of making a true, though not perhaps the shortest and easiest demonstration.

The rules of demonstration are but of two kinds: one, that the principles be true and evident definitions; the other, that the inferences be necessary. And of true and evident definitions, the best are those which declare the cause or generation of that subject, whereof the proper passions are to be demonstrated. For science is that knowledge which is derived from the comprehension of the cause. But when the cause appeareth not, then may, or rather must we define some known property of the subject, and from thence derive some possible way, or ways, of the generation. And the more ways of generation are explicated, the more easy will be the derivation of the properties; whereof some are more immediate to one, some to another generation. He therefore that proceedeth from untrue, or not understood definitions, is ignorant of that he goes about; which is an ill-favoured fault, be the matter he undertaketh easy or difficult, because he was not forced to undergo a greater charge than he could carry through. But he that from right definitions maketh a false conclusion, erreth through human frailty, as being less awake, more troubled with other thoughts, or more in haste when he was in writing. Such faults, unless they be very frequent, are not attended with shame, as being common to all men, or are at least less ugly than the former, except then, when he that committeth them reprehendeth the same in other men. For that is in every man intolerable, which he cannot tolerate in another. But to the end that the faults of both kinds may by every man be well understood, it will not be amiss to examine them by some such demonstrations as are publicly extant. And for this purpose I will take such as are in mine and in your books, and begin with your Elenchus of the geometry contained in my book De Corpore; to which I will also join your book lately set forth concerning the Angle of Contact, Conic Sections , and your Arithmetica Infinitorum; and then examine the rest of my philosophy, and yours that oppugn it. For I will take leave to consider you both everywhere as one author, because you publicly declare your approbation of one another’s doctrine.

My first definition is of a line, of length, and of a point. “The way,” say I, “of a body moved, in which magnitude (though it always have some magnitude) is not considered, is called a line; and the space gone over by that motion, length, or one and a simple dimension.” To this definition you say, first, “what mathematician did ever thus define a line or length?” Whether you call in others for help or testimony, it is not done like a geometrician; for they use not to prove their conclusions by witnesses, but rely upon the strength of their own reason; and when your witnesses appear, they will not take your part. Secondly, you grant that what I say is true, but not a definition. But to tell you truly what it is which we call a line, is to define a line. Why then is not this a definition? “Because,” say you in the first place, “it is not a reciprocal proposition.” But by your favour it is reciprocal. For not only the way of a body whose quantity is not considered is a line, but also every line is, or may be conceived to be, the way of a body so moved. And if you object that there is a difference between is and may be conceived to be, Euclid, whom you call to your aid, will be against you in the fourteenth definition before his eleventh Element; where he defines a sphere just as convertibly as I define a line; except you think the globes of the sun and stars cannot be globes, unless they were made by the circumduction of a semicircle; and again in the eighteenth definition, which is of a cone, unless you admit no figure for a cone, which is not generated by the revolution of a triangle; and again, in the twentieth definition, which is of a cylinder, except it be generated by the circumvolution of a parallelogram. Euclid saw that what proper passion soever should be derived from these his definitions, would be true of any other cylinder, sphere, or cone, though it were otherwise generated; and the description of the generation of any one being by the imagination applicable to all, which is equivalent to convertible, he did not believe that any rational man could be misled by learning logic to be offended with it. Therefore this exception proceedeth from want of understanding, that is, from ignorance of the nature, and use of a definition.

Again, you object and ask: “What need is there of motion, or of body moved, to make a man understand what is a line? Are not lines in a body at rest, as well as in a body moved? And is not the distance of two resting points length, as well as the measure of the passage? Is not length one and a simple dimension, and one and a simple dimension line? Why then is not line and length all one?” See how impertinent these questions are. Euclid defines a sphere to be a solid figure described by the revolution of a semicircle about the unmoved diameter. Why do you not ask, what need there is to the understanding of what a sphere is, to bring in the motion of a semicircle? Is not a sphere to be understood without such motion? Is not the figure so made a sphere without this motion? And where he defines the axis of a sphere to be that unmoved diameter, may not you ask, whether there be no axis of a sphere, when the whole sphere, diameter and all, is in motion? But it is not to my purpose to defend my definition by the example of that of Euclid. Therefore first, I say, to me, howsoever it may be to others, it was fit to define a line by motion. For the generation of a line is the motion that describes it. And having defined philosophy in the beginning, to be the knowledge of the properties from the generation, it was fit to define it by its generation. And to your question, is not distance length? I answer, that though sometimes distance be equivalent to length, yet certainly the distance between the two ends of a thread wound up into a clue is not the length of the thread; for the length of the thread is equal to all the windings whereof the clue is made. But if you will needs have distance and length to be all one, tell me of what the distance between any two points is the length. Is it not the length of the way? And how is that called way, which is not defined by some motion? And have not several ways between the same places, as by land and by water, several lengths? But they have but one distance, because the distance is the shortest way. Therefore between the length of the path, and the distance of places, there is a real difference in this case, and in all cases a difference of the consideration. Your objection, that line is longitude, proceeds from want of understanding English. Do men ever ask what is the line of a thread, or the line of a table, or of any other body? Do they not always ask what is the length of it? And why, but because they use their own judgments, not yet corrupted by the subtlety of mistaken professors. Euclid defines a line to be length without breadth. If those terms be all one, why said he not that a line is a line without breadth? But what definition of a line give you? None. Be contented then with such as you receive, and with this of mine, which you shall presently see is not amiss.

Your next objections are to my definition of a point. Which definition adhereth to the former in these words, “and the body itself is called a point.” Here again you call for help: “Quis unquum mortalium, etc. What mortal man, what sober man, did ever so define a point?” It is well, and I take it to be an honour to be the first that do so. But what objection do you bring against it. This: “That a point added to a point, if it have magnitude, makes it greater.” I say it doth so, but then presently it loseth the name of a point, which name was given to signify that it was not the meaning of him that used it in demonstration to add, subtract, multiply, divide, or any way compute it. Then you come in with, “perhaps you will say though it have magnitude, that magnitude is not considered.” You need not say perhaps. You know I affirm it; and therefore your argument might have been left out, but that it gave you an occasion of a digression into scurvy language.

And whereas you ask why I defined not a point thus: “Punctum est corpus quod non consideratur esse corpus, et magnum quod non consideratur esse magnum.” I will tell you why. First, because it is not Latin. Secondly, because when I had defined it by corpus, there was no need to define it again by magnum. I understand very well this language, “punctum est corpus, quod non consideratur ut corpus.” A point is a body not considered as body. But punctum est corpus, quod non consideratur esse corpus, vel esse magnum, is not Latin; nor the version of it, a point is a body which is not considered to be a body, English. My definition was, that a point is that body whose magnitude is not considered, not reckoned, not put to account in demonstration. And I exemplified the same by the body of the earth describing the ecliptic line; because the magnitude is not there reckoned nor chargeth the ecliptic line with any breadth. But I perceive you understand not what the word consideration signifieth, but take it for comparison or relation; and say I ought to define a point simply, and not by relation to a great body; as if to reckon and to compare were the same thing. “Omnia mihi,” saith Cicero, “provisa et considerata sunt.” I have provided and reckoned everything. There is a great difference between reckoning and relation.

Again, you ask, why corpus motum, a body moved? I will tell you; because the motion was necessary for the generation of a line. And though after the generation of the line the point should rest, yet it is not necessary from this definition that it should be no more a point; nor when Euclid defines a sphere by the circumduction of a semicircle upon an axis that resteth, doth it follow from thence when the sphere, axis, centre and all, as that of the earth, is moved from place to place, that it is no more an axis.

Lastly, you object “that motion is accidentary to a point, and consequently not essential, nor to be put into the definition.” And is not the circumduction of a semicircle accidentary to a sphere? Or do you think the sphere of the sun was generated by the revolution of a semicircle? And yet it was thought no fault in Euclid to put the motion into the definition of a sphere.

The conceit you have concerning definitions, that they must explicate the essence of the thing defined, and must consist of a genus and a difference, is not so universally true as you are made believe, or else there be very many insufficient definitions that pass for good with you in Euclid. You are much deceived if you think these woful notions of yours, and the language that doth everywhere accompany them, show handsomely together. Or that such grounds as these be able to sustain so many, and so haughty reproaches as you advance upon them, so as they fall not, as you shall see immediately, upon your own head. I say a point hath quantity, but not to be reckoned in demonstrating the properties of lines, solids, or superficies; you say it hath no quantity at all, but is plainly nothing.

The first of the petitions of Euclid is, “that a line may be drawn from point to point at any distance.” The second, “that a straight line may be produced.” The third, “that on any centre a circle may be described at any distance.” And the eighth axiom (which Sir H. Savile observes to be the foundation of all geometry) is this, “Quæ sibi mutuo congruunt, etc. Those things that are applied to one another in all points are equal.” All or any of these principles being taken away, there is not in Euclid one proposition demonstrated or demonstrable. If a point have no quantity, a line can have no latitude; and because a line is not drawn but by motion, by motion of a body, and body imprinteth latitude all the way, it is impossible to draw or produce a straight line, or to describe a circular line without latitude. Also if a line have no latitude, one straight line cannot be applied to another. To them therefore that deny a point to have quantity, that is, a line to have latitude, the forenamed principles are not possible, and consequently no proposition in geometry is demonstrated or demonstrable. You therefore that deny a point to have quantity, and a line to have breadth, have nothing at all of the science of geometry. The practice you may have, but so hath any man that hath learned the bare propositions by heart; but they are not fit to be professors either of geometry or of any other science that dependeth on it. Some man perhaps may say that this controversy is not much worth, and that we both mean the same thing. But that man, though in other things prudent enough, knoweth little of science and demonstration. For definitions are not only used to give us the notions of those things whose appellations are defined, for many times they that have no science have the ideas of things more perfect than such as are raised by definitions. As who is there that understandeth not better what a straight line is, or what proportion is, and what many other things are, without definition, than some that set down the definitions of them. But their use is, when they are truly and clearly made, to draw arguments from them for the conclusions to be proved. And therefore you that in your following censures of my geometry, take your argument so often from this, that a point is nothing, and so often revile me for the contrary, are not to be allowed such an excuse. He that is here mistaken, is not to be called negligent in his expression, but ignorant of the science.

In the next place, you take exceptions to my definition of equal bodies, which is this: “Corpora æqualia sunt quæ eundem locum possidere possunt. Equal bodies are those which may have the same place.” To which you object impertinently, that I may as well define a man to be, he that may be prince of Transylvania, wittily, as you count wit. Formerly in every definition, you exacted an explication of the essence. You are therefore of opinion that the possibility of being prince of Transylvania is no less essential to a man, than the possibility of the being of two bodies successively in the same place, is essential to bodies equal.

You take no notice of the twenty-third article of this same chapter, where I define what it is we call essence, namely, that accident for which we give the thing its name. As the essence of a man is his capacity of reasoning; the essence of a white body, whiteness, &c., because we give the name of man to such bodies as are capable of reasoning, for that their capacity; and the name of white to such bodies as have that colour, for that colour. Let us now examine why it is that men say bodies are one to another equal; and thereby we shall be able to determine whether the possibility of having the same place be essential or not to bodies equal, and consequently whether this definition be so like to the defining of a man by the possibility of being prince of Transylvania as you say it is. There is no man, besides such egregious geometricians as yourselves, that inquireth the equality of two bodies, but by measure. And for liquid bodies, or the aggregates of innumerable small bodies, men (men, I say) measure them by putting them one after another into the same vessel, that is to say, into the same place, as Aristotle defines place, or into the space determined by the vessel, as I define place. And the bodies that so fill the vessel, they acknowledge and receive for equal. But though, when hard bodies cannot be so measured, without the incommodity or trouble of altering their figure, they then enquire, if the bodies are both of the same kind, their equality by weight, knowing, without your teaching, that equal bodies of the same nature weigh proportionably to their magnitudes; yet they do it not for fear of missing of the equality, but to avoid inconvenience or trouble. But you (you, I say), that have no definition of equals, neither received from others, nor framed by yourselves, out of your shallow meditation and deep conceit of your own wits, contend against the common light of nature. So much is unheedy learning a hinderance to the knowledge of the truth, and changeth into elves those that were beginning to be men.

Again, when men inquire the equality of two bodies in length, they measure them by a common measure; in which measure they consider neither breadth nor thickness, but how the length of it agreeth, first with the length of one of the bodies, then with the length of the other. And both the bodies whose lengths are measured, are successively in the same place under their common measure. Place therefore in lines also, is the proper index and discoverer of equality and inequality. And as in length, so it is in breadth and thickness, which are but lengths otherwise taken in the same solid body. But now when we come from this equality and inequality of lengths known by measure, to determine the proportions of superficies and of solids, by ratiocination, then it is that we enter into geometry; for the making of definitions, in whatsoever science they are to be used, is that which we call philosophia prima. It is not the work of a geometrician, as a geometrician, to define what is equality, or proportion, or any other word he useth, though it be the work of the same man, as a man. His geometrical part is, to draw from them as many true and useful theorems as he can.

You object secondly, that a pyramis may be equal to a cube whilst it is a pyramis. True. And so also whilst it is a pyramis it hath a possibility by flexion and transposition of parts to become a cube, and to be put into the place where another cube equal to it was before. This is to argue like a child that hath not yet the perfect understanding of any language.

In the third and fourth objection, you teach me to define equal bodies (if I will needs define them by place) by the equality of place, and to say, that bodies are equal that have equal places. Teach others, if you can, to measure their grain, not by the same, but equal bushels.

In the fifth objection, you except against the the word can, in that I say that bodies are equal which can fill the same place. For the greater body can, you say, fill the place of the less, though not reciprocally the less of the greater. It is true, that though the place of the less can never be the place of the greater, yet it may be filled by a part of the greater. But it is not then the greater body that filleth the place of the less, but a part of it, that is to say, a less body. Howsoever, to take away from simple men this straw they stumble at, I have now put the definition of equal bodies into these words: equal bodies are those whereof every one can fill the place of every other. And if my definition displease you, propound your own, either of equal bodies, or of equals simply. But you have none. Take therefore this of mine.

The sixth is a very admirable exception. “What,” say you, “if the same body can sometimes take up a greater, sometimes a lesser place, as by rarefaction and condensation?” I understand very well that bodies may be sometimes thin and sometimes thick, as they chance to stand closer together or further from one another. So in the mathematic schools, when you read your learned lectures, you have a thick or thronging audience of disciples, which in a great church would be but a very thin company. I understand how thick and thin may be attributed to bodies in the plural, as to a company; but I understand not how any one of them is thicker in the school than in the church; or how any one of them taketh up a greater room in the school, when he can get in, than in the street. For I conceive the dimensions of the body, and of the place, whether the place be filled with gold or with air, to be coincident and the same; and consequently both the quantity of the air, and the quantity of the gold, to be severally equal to the quantity of the place; and therefore also, by the first axiom of Euclid, equal to one another; insomuch as if the same air should be by condensation contained in a part of the place it had, the dimensions of it would be the same with the dimensions of part of the place, that is, should be less than they were, and by consequence the quantity less. And then either the same body must be less also, or we must make a difference between greater bodies and bodies of greater quantity; which no man doth that hath not lost his wits by trusting them with absurd teachers. When you receive salary, if the steward give you for every shilling a piece of sixpence, and then say, every shilling is condensed into the room of sixpence, I believe you would like this doctrine of yours much the worse. You see how by your ignorance you confound the affairs of mankind, as far forth as they give credit to your opinions, though it be but little. For nature abhors even empty words, such as are (in the meaning you assign them), rarefying and condensing. And you would be as well understood if you should say (coining words by your own power), that the same body might take up sometimes a greater, sometimes a lesser place, by wallifaction and wardensation, as by rarefaction and condensation. You see how admirable this your objection is.

In the seventh objection you bewray another kind of ignorance, which is the ignorance of what are the proper works of the several parts of philosophy. “Though it were out of doubt,” say you, “that the same body cannot have several magnitudes, yet seeing it is matter of natural philosophy, nor hath anything to do with the present business, to what purpose is it to mention it in a mathematical definition?” It seems by this, that all this while you think it is a piece of the geometry of Euclid, no less to make the definitions he useth, than to infer from them the theorems he demonstrateth. Which is not true. For he that telleth you in what sense you are to take the appellations of those things which he nameth in his discourse, teacheth you but his language, that afterwards he may teach you his art. But teaching of language is not mathematic, nor logic, nor physic, nor any other science; and therefore to call a definition, as you do, mathematical, or physical, is a mark of ignorance, in a professor inexcusable. All doctrine begins at the understanding of words, and proceeds by reasoning till it conclude in science. He that will learn geometry must understand the terms before he begin, which that he may do, the master demonstrateth nothing, but useth his natural prudence only, as all men do when they endeavour to make their meaning clearly known. For words understood are but the seed, and no part of the harvest of philosophy. And this seed was it, which Aristotle went about to sow in his twelve books of metaphysics, and in his eight books concerning the hearing of natural philosophy. And in these books he defineth time, place, substance or essence, quantity, relation, &c., that from thence might be taken the definitions of the most general words for principles in the several parts of science. So that all definitions proceed from common understanding; of which, if any man rightly write, he may properly call his writing philosophia prima, that is, the seeds, or the grounds of philosophy. And this is the method I have used, defining place, magnitude, and the other the most general appellations in that part which I entitle philosophia prima. But you now, not understanding this, talk of mathematical definitions. You will say perhaps that others do the same as well as you. It may be so. But the appeaching of others does not make your ignorance the less.

In the eighth place you object not, but ask me why I define equal bodies apart? I will tell you. Because all other things which are said to be equal, are said to be so from the equality of bodies; as two lines are said to be equal, when they be coincident with the length of one and the same body; and equal times, which are measured by equal lengths of body, by the same motion. And the reason is, because there is no subject of quantity, or of equality, or of any other accident but body; all which I thought certainly was evident enough to any uncorrupted judgment; and therefore that I needed first to define equality in the subject thereof, which is body, and then to declare in what sense it was attributed to time, motion, and other things that are not body.

The ninth objection is an egregious cavil. Having set down the definition of equal bodies, I considered that some men might not allow the attribute of equality to any things but those which are the subjects of quantity, because there is no equality, but in respect of quantity. And to speak rigidly, magnum et magnitudo are not the same thing; for that which is great, is properly a body, whereof greatness is an accident. In what sense therefore, might you object, can an accident have quantity? For their sakes therefore that have not judgment enough to perceive in what sense men say the length is so long, or the superficies so broad, &c. I added these words: “Eadem ratione (qua scilicet corpora dicuntur æqualia) magnitudo magnitudini æqualis dicitur,” that is, in the same manner, as bodies are said to be equal, their magnitudes also are said to be equal. Which is no more than to say, when bodies are equal, their magnitudes also are called equal. When bodies are equal in length, their lengths are also called equal. And when bodies are equal in superficies, their superficies are also called equal. All which is common speech, as well amongst mathematicians, as amongst common people; and, though improper, cannot be altered, nor needeth to be altered to intelligent men. Nevertheless I did think fit to put in that clause, that men might know what it is we call equality, as well in magnitudes as in magnis, that is, in bodies. Which you so interpret, as if it bore this sense, that when bodies are equal their superficies also must be equal, contrary to your own knowledge, only to take hold of a new occasion of reviling. How unhandsome and unmanly this is, I leave to be judged by any reader that hath had the fortune to see the world, and converse with honest men.

Against the fourteenth article, where I prove that the same body hath always the same magnitude, you object nothing but this, that though it be granted, that the same body hath the same magnitude, while it resteth, yet I bring nothing to prove that when it changeth place, it may not also change its magnitude by being enlarged or contracted. There is no doubt but to a body, whether at rest or in motion, more body may be added, or part of it taken away. But then it is not the same body, unless the whole and the part be all one. If the schools had not set your wit awry, you could never have been so stupid as not to see the weakness of such objections. That which you add in the end of your objections to this eighth chapter, that I allow not Euclid this axiom gratis, that the whole is greater than a part, you know to be untrue.

At my eleventh chapter, you enter into dispute with me about the nature of proportion. Upon the truth of your doctrine therein, and partly upon the truth of your opinions concerning the definitions of a point, and of a line, dependeth the question whether you have any geometry or none; and the truth of all the demonstrations you have in your other books, namely of the Angle of Contact , and Arithmetica Infinitorum. Here I say you enter, how you will get out, your reputation saved, we shall see hereafter.

When a man asketh what proportion one quantity hath to another, he asketh how great or how little the one is comparatively to, or in respect of the other. When a geometrician prefixeth before his demonstrations a definition, he doth it not as a part of his geometry, but of natural evidence, not to be demonstrated by argument, but to be understood in understanding the language wherein it is set down; though the matter may nevertheless, if besides geometry he have wit, be of some help to his disciple to make him understand it the sooner. But when there is no significant definition prefixed, as in this case, where Euclid’s definition of proportion, that it is a whatshicalt habitude of two quantities, &c., is insignificant, and you allege no other, every one that will learn geometry, must gather the definition from observing how the word to be defined is most constantly used in common speech. But in common speech if a man shall ask how much, for example, is six in respect of four, and one man answer that it is greater by two, and another that it is greater by half of four, or by a third of six, he that asked the question will be satisfied by one of them, though perhaps by one of them now, and by the other another time, as being the only man that knoweth why he himself did ask the question. But if a man should answer, as you would do, that the proportion of six to two is of those numbers a certain quotient, he would receive but little satisfaction. Between the said answers to this question, how much is six in respect of four? there is this difference. He that answereth that it is more by two, compareth not two with four, nor with six, for two is the name of a quantity absolute. But he that answereth it is more by half of four, or by a third of six, compareth the difference with one of the differing quantities. For halfs and thirds, &c. are names of quantity compared.

From hence there ariseth two species or kinds of (ratio) proportion, into which the general word proportion may be divided. The one whereof, namely, that wherein the difference is not compared with either of the differing quantities, is called ratio arithmetica, arithmetical proportion; the other ratio geometrica, geometrical proportion; and, because this latter is only taken notice of by the name of proportion, simply proportion. Having considered this, I defined proportion,