Chapter X (2)

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, chapter II. article 3, in this manner: “_Ratio est relatio antecedentis ad consequens secundum magnitudinem_:” _Proportion is the relation of the antecedent to the consequent in magnitude_; having immediately before defined relatives, antecedent, and consequent, in the same article, and by way of explication added, that such relation was nothing else but that one of the quantities was equal to the other, or exceeded it by some quantity, or was by some quantity exceeded by it. And for exemplification of the same, I added further, that the proportion of three to two was, that three exceeded two by a unity; but said not that the unity, or the difference whatsoever it were, was their proportion, _for unity, and to exceed another by unity_, is not the same thing. This is clear enough to others; let us therefore see why it is not so to you. You say I make proportion to consist in that which remaineth after the lesser quantity is subtracted out of the greater; and that you make it to consist in the quotient, when one number is divided by the other. Wherein you are mistaken; first, in that you say, I make the proportion to consist in the remainder. For I make it to consist in the act of exceeding, or of being exceeded, or of being equal; whereas the remainder is always an absolute quantity, and never a proportion. To be more or less than another number by two, is not the number two; likewise to be equal to two, where the difference is _nothing_, is not that _nothing_? Again, you mistake in saying the proportion consisteth in the quotient. For divide twenty by five, the quotient is four. Is it not absurd to say that the proportion of five to twenty, or of twenty to five, is four? You may say the proportion of five to twenty, is the proportion of one to four. And so say I. And you may therefore also say, that the proportion of one to four is a measure of the proportion of five to twenty, as being equal. And so say I. But that is only in geometrical proportion, and not in proportion universally. For though the _species_ obtain the denomination of the genus, yet it is not the _genus_. But as the quotient giveth us a measure of the proportion of the dividend to the divisor in geometrical proportion, so also the remainder after subtraction is the measure of proportion arithmetical.

You object in the next place, “that if the proportion of one quantity to another be nothing but the excess or defect, then, wheresoever the excess or defect is the same, there the proportion is the same.” This you say follows in your logic, and from thence, that the proportion of three to two, and five to four is the same. But is not three to two, and five to four, where the excess is the same number, the same proportion arithmetical? And is not arithmetical proportion, proportion? You take here (_ratio_) proportion, which is the _genus_, for that _species_ of it which is called geometrical, because usually this species has the name of proportion simply. Also that the proportion of three to two, is the same with that of nine to six; is it not because the excesses are one and three, the same portions of three and nine, that is to say the same excesses comparatively? I wonder you ask me not what is the _genus_ of arithmetical and geometrical proportions, and what the _difference_? The _genus_ is (_ratio_) proportion, or comparison in magnitude, and the _difference_ is that one comparison is made by the absolute quantity, the other by the comparative quantity, of the excess or defect, if there be any. Can anything be clearer than this? You after come in with _ignosce habitudini_ to no purpose. I am not so inhuman as not to pardon dulness or madness: they are not voluntary faults. But when men adventure voluntarily to talk of that they understand not censoriously and scornfully, I may tell them of it.

This difference between the excesses or defects, as they are simply or comparatively reckoned, being thus explained, all the rest of that you say in your objections to this eleventh chapter (saving that art. 5 for _ratio binarii ad quinarium est superari ternario_, as it is in other places, I have put too hastily _ratio binarii ad quinarium est ternarius_), will be understood by every reader to be frivolous, and to proceed from the ignorance of what proportion is.

At the twelfth chapter you only note that I say, _that the proportion of inequality is quantity, but the proportion of equality not quantity_, and refer that which you have to say against it to the chapter following; to which place I shall also come in the following lesson.

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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 III.

You begin your reprehension of my thirteenth chapter with a question; whereas _I_ divide proportion into arithmetical and geometrical. You ask me what _proportion it is I so divide_. Euclid divides an angle into right, obtuse, and acute. I may ask you as pertinently, what angle it is he so divides? Or, when you divide _animal_ into _homo_ and _brutum_, what animal that is, which you so divide? You see by this, how absurd your question is. But you say the definition of proportion which I make at Chap. II. art. 3., namely, that proportion is the comparison of two magnitudes, one to another, agreeth not, neither with arithmetical, nor with geometrical proportion. I believe you thought so then, but having read what I have said in the end of the last lesson, if you think so still, your fault will be too great to be pardoned easily. But why did you think so before? Is it not because there was no definition in Euclid of proportion universal, and because he maketh no mention of proportion arithmetical, and because you had not in your minds a sufficient notion thereof yourselves to supply that defect? And is not this the cause also, why you put in this parenthesis (if arithmetical proportion ought to be called proportion)? Which is a confession that you know not whether there be such a thing as arithmetical proportion or not, notwithstanding that on all occasions you speak of arithmetical proportionals. Yes, this was it that made you think that proportion universally, and proportion geometrical, is the same, and yet to say you cannot tell whether they be the same or not. It is no wonder, therefore, if in such confusion of the understanding, you apprehend not that the proportions of two to five, and nine to twelve, are the same; so you are blinded by seeing that they are not the same proportions geometrical. Nor doth it help you that I say the difference is the proportion; for by difference you might, if you would, have understood the act of differing.

At the second article you note for a fault in method, that _after I had used the words antecedent and consequent of a proportion in some of the precedent chapters, I define them afterwards_. I do not believe you say this against your knowledge, but that the eagerness of your malice made you oversee; therefore go back again to the third article of chapter II. where, having defined correlatives, I add these words, _of which the first is called the_ antecedent, _the second the_ consequent. This is but an oversight, though such as in me you would not have excused.