Chapter X (7)

And first, how came it into your minds that a man can be an atheist, I mean an atheist in his conscience? I know that David confesseth of himself, upon sight of the prosperity of the wicked, that his feet had almost slipped, that is, that he had slipped into a short doubtfulness of the Divine Providence. And if anything else can cause a man to slip in the same kind, it is the seeing such as you (who though you write nothing but what is dictated to each of you by a doctor of divinity) do break the greatest of God’s commandments, which is charity, in every line before his face. And though such forgettings of God be somewhat more than short doubtings, and sudden transportations incident to human passion, yet I do not for that cause think you atheists and enemies of religion, but only ignorant and imprudent Christians. But how, I say, could you think me an atheist, unless it were because finding your doubts of the Deity more frequent than other men do, you are thereby the apter to fall upon that kind of reproach? Wherein you are like women of poor and evil education when they scold; amongst whom the readiest disgraceful word is whore: why not thief, or any other ill name, but because, when they remember themselves, they think that reproach the likeliest to be true?

Secondly, tell me what crime it was which the Latins called by the name of _scelus_? You think not, unless you be Stoics, that all crimes are equal. _Scelus_ was never used but for a crime of greatest mischief, as the taking away of life and honour; and besides, basely acted, as by some clandestine way, or by such a way as might be covered with a lie. But when you insinuate in a writing published that I am an atheist, you make yourselves authors to the multitude, and do all you can to stir them up to attempt upon my life; and if it succeed, then to sneak out of it by leaving the fault on them that are but actors. This is to endeavour great mischief basely, and therefore _scelus_. Again, to deprive a man of the honour he hath merited, is no little wickedness; and this you endeavour to do by publishing falsely that I challenge as my own the inventions of other men. This is therefore _scelus_ publicly to tell all the world that I will be angry with all men that do not presently submit to my dictates; to deprive me of the friendship of all the world; great damage, and a lie, and yours. For to publish any untruth of another man to his disgrace, on hearsay from his enemy, is the same fault as if he published it on his own credit. If I should say I have heard that Dr. Wallis was esteemed at Oxford for a simple fellow, and much inferior to his fellow-professor Dr. Ward (as indeed I have heard, but do not believe it), though this be no great disgrace to Dr. Wallis, yet he would think I did him injury. Therefore public accusation upon hearsay is _scelus_. And whosoever does any of these things does _sceleratè_. But you the professors of the mathematics at Oxford, by the advice of two doctors of divinity have dealt thus with me. Therefore you have done, I say not foolishly, though no wickedness be without folly, but _sceleratè_, ὅπερ ἔδει δεῖξαι.

Thirdly, it is ill manners, in reprehending truth, to send a man in a boasting way to your own errors; as you the professor of geometry have often sent me to your two tractates of the _Angle of Contact_ and _Arithmetica Infinitorum_.

Fourthly, it is ill manners, to diminish the just reputation of worthy men after they be dead, as you the professor of geometry have done in the case of Joseph Scaliger.

Fifthly, when I had in my _Leviathan_ suffered the clergy of the Church of England to escape, you did imprudently in bringing any of them in again. An Ulysses upon so light an occasion would not have ventured to return again into the cave of Polyphemus.

Lastly, how ill does such levity and scurrility, which both of you have shown so often in your writings, become the gravity and sanctity requisite to the calling of the ministry? They are too many to be repeated. Do but consider, you the geometrician, how unhandsome it is to play upon my name, when both yours and mine are plebeian names; though from Willis by Wallis, you go from yours in Wallisius. The jest of using at every word _mi Hobbi_, is lost to them beyond sea. But this is not so ill as some of the rest. I will write out one of them, as it is in the fourth page of your _Elenchus_: “_Whence it appears that your Empusa was of the number of those fairies which you call in English hob-goblins. The word is made of_ ἕν and πους; _and thence comes the children’s play called the play of Empusa, Anglicè_ (hitherto in Latin all but _hob-goblins_, then follows in English) _fox, fox, come out of your hole_ (then in Latin again), _in which the boy that is called the fox, holds up one foot, and jumps with the other, which in English is to hop_.” When a stranger shall read this, and hoping to find therein some witty conceit, shall with much ado have gotten it interpreted and explained to him, what will he think of our doctors of divinity at Oxford, that will take so much pains as to go out of the language they set forth in, for so ridiculous a purpose? You will say it is a pretty _paranomasia_. How you call it there I know not, but it is commonly called here a _clinch_; and such a one as is too insipid for a boy of twelve years old, and very unfit for the sanctity of a minister, and gravity of a doctor of divinity. But I pray you tell me where it was you read the word _empusa_ for the boy’s play you speak of, or for any other play amongst the Greeks? In this (as you have done throughout all your other writings) you presume too much upon your first cogitations. There be a hundred other scoffing passages, and ill-favoured attributes given me in both your writings, which the reader will observe without my pointing to them, as easily as you would have him; and which perhaps some young students, finding them full of gall, will mistake for salt. Therefore to disabuse those young men, and to the end they may not admire such kind of wit, I have here and there been a little sharper with you than else I would have been. If you think I did not spare you, but that I had not wit enough to give you as scornful names as you give me, are you content I should try? Yes (you the geometrician will say) give me what names you please, so you call me not _Arithmetica Infinitorum_. I will not. Nor _Angle of Contact_ ; nor _Arch Spiral_ ; nor _Quotient_ . I will not. But I here dismiss you both together. So go your ways, you _Uncivil Ecclesiastics, Inhuman Divines, Dedoctors of morality, Unasinous Colleagues, Egregious pair of Issachars, most wretched Vindices and Indices Academiarum_; and remember Vespasian’s law, that it is uncivil to give ill language first, but civil and lawful to return it. But much more remember the law of God, to obey your sovereigns in all things; and not only not to derogate from them, but also to pray for them, and as far as you can to maintain their authority, and therein your own protection. And, do you hear? take heed of speaking your mind so clearly in answering my _Leviathan_, as I have done in writing it. You should do best not to meddle with it at all, because it is undertaken, and in part published already, and will be better performed, from term to term, by one Christopher Pike.

ΣΤΙΓΜΑΙ

Αγεωμετρίας, Αγροικίας, Αντίπολιτείας, Αμαθείας,

OR

MARKS

OF THE

ABSURD GEOMETRY, RURAL LANGUAGE, SCOTTISH
CHURCH POLITICS, AND BARBARISMS

OF

JOHN WALLIS,
PROFESSOR OF GEOMETRY AND DOCTOR OF DIVINITY.

BY

THOMAS HOBBES,

OF MALMESBURY.

TO THE RIGHT HONOURABLE

HENRY, LORD PIERREPONT,

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

==========

MY MOST NOBLE LORD,

I did not intend to trouble your Lordship twice with this contention between me and Dr. Wallis. But your Lordship sees how I am constrained to it; which, whatsoever reply the Doctor makes, I shall be constrained to no more. That which I have now said of his Geometry, Manners, Divinity, and Grammar, altogether is not much, though enough. As for that which I here have written concerning his Geometry, which you will look for first, is so clear, that not only your Lordship, and such as have proceeded far in that science, but also any man else that doth but know how to add and subtract proportions, (which is taught at the twenty-third proposition of the sixth of Euclid), may see the Doctor is in the wrong. That which I say of his ill language and politics is yet shorter. The rest, which concerneth grammar, is almost all another man’s, but so full of learning of that kind, as no man that taketh delight in knowing the proprieties of the Greek and Latin tongues, will think his time ill bestowed in the reading it. I give the Doctor no more ill words, but am returned from his manners to my own. Your Lordship may perhaps say, my compliment in my title-page is somewhat coarse; and it is true. But, my Lord, it is since the writing of the title-page, that I am returned from the Doctor’s manners to my own; which are such as I hope you will not be ashamed to own me, my Lord, for one of

Your Lordship’s most humble

and obedient servants,

THOMAS HOBBES.

==========

TO

DOCTOR WALLIS,

IN ANSWER TO HIS

SCHOOL DISCIPLINE

---

SIR,

When unprovoked you addressed unto me, in your _Elenchus_, your harsh compliment with great security, wantonly to show your wit, I confess you made me angry, and willing to put you into a better way of considering your own forces, and to move you a little as you had moved me, which I perceive my lessons to you have in some measure done; but here you shall see how easily I can bear your reproaches, now they proceed from anger, and how calmly I can argue with you about your geometry and other parts of learning.

I shall in the first part confer with you about your _Arithmetica Infinitorum_, and afterwards compare our manner of elocution; then your politics; and last of all your grammar and critics. Your spiral line is condemned by him whose authority you use to prove me a plagiary, (that is, a man that stealeth other men’s inventions, and arrogates them to himself), whether it be Roberval or not that writ that paper, I am not certain. But I think I shall be shortly; but whosoever it be, his authority will serve no less to show that your doctrine of the spiral line, from the fifth to the eighteenth proposition of your _Arithmetica Infinitorum_, is all false; and that the principal fault therein (if all faults be not principal in geometry, when they proceed from ignorance of the science) is the same that I objected to you in my _Lessons_. And for the author of that paper, when I am certain who it is, it will be then time enough to vindicate myself concerning that name of plagiary. And whereas he challenges the invention of your method delivered in your _Arithmetica Infinitorum_, to have been his before it was yours, I shall, I think, by and by say that which shall make him ashamed to own it; and those that writ those encomiastic epistles to you ashamed of the honour they meant to you. I pass therefore to the nineteenth proposition, which in Latin is this: your geometry!

“_Si proponatur series quantitatum in duplicata ratione arithmetice proportionalium (sive juxta seriem numerorum quadraticorum) continue crescentium, a puncto vel 0 inchoatarum, (puta ut 0. 1. 4. 9. 16. etc.), propositum sit, inquirere quam habeat illa rationem ad seriem totidem maximæ æqualium._

“_Fiat investigatio per modum inductionis ut_ (_in prop. 1_)

_Eritque_,

(0 + 1 = 1)/(1 + 1 = 2) = (1)/(3) + (1)/(6)

(0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (1)/(3) + (1)/(12)

(0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) = (1)/(3) + (1)/(18) _et sic deinceps_.

“_Ratio proveniens est ubique major quam subtripla seu (1)/(3); excessus autem perpetuo decrescit prout numerus terminorum augetur (puta (1)/(6) (1)/(12) (1)/(18) (1)/(24) etc.) aucto nimirum fractionis denominatore sive consequente rationis in singulis locis numero senario (ut patet) ut sit rationis provenientis excessus supra subtriplam, ea quam habet unitas ad sextuplum numeri terminorum post 0; adeoque._”

That is, if there be propounded a row of quantities in duplicate proportion of the quantities arithmetically proportional (or proceeding in the order of the square numbers) continually increasing; and beginning at a point or 0; let it be propounded to find what proportion the row hath; to as many quantities equal to the greatest;

Let it be sought by induction (as in the first proposition).

The proportion arising is everywhere greater than subtriple, or (1)/(3), and the excess perpetually decreaseth as the number of terms is augmented, as here, (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. denominator of the fraction being in every place augmented by the number six, as is manifest; so that the excess of the rising proportion above subtriple is the same which unity hath to six times the number of terms after 0; and so.

Sir, in these your characters I understand by the cross + that the quantities on each side of it are to be added together and make one aggregate; and I understand by the two parallel lines = that the quantities between which they are placed are one to another equal; this is your meaning, or you should have told us what you meant else; I understand also, that in the first row 0 + 1 is equal to 1, and 1 + 1 equal to 2; and that in the second row 0 + 1 + 4 is equal to 5; and 4 + 4 + 4 equal to 12; but (which you are too apt to grant) I understand your symbols no further; but must confer with yourself about the rest.

And first I ask you (because fractions are commonly written in that manner) whether in the uppermost row (which is (0 + 1 = 1)/(1 + 1 = 2) = (1)/(3) + (1)/(6))(0)/(1) be a fraction, (1)/(1) be a fraction, (1)/(2) be a fraction, that is to say, a part of an unit, and if you will, for the cypher’s sake, whether (0)/(1), be an infinitely little part of 1; and whether (1)/(1) or 1 divided by 1 signify an unity? if that be your meaning, then the fraction (0)/(1) added to the fraction (1)/(1) is equal to the fraction (1)/(2): But the fraction (0)/(1) is equal to O; therefore the fraction (0)/(1) + (1)/(1) is equal to the fraction (1)/(1); and (1)/(1) equal to (1)/(2) which you will confess to be an absurd conclusion, and cannot own that meaning.

I ask you therefore again, if by (0)/(1) you mean the proportion of 0 to 1; and consequently by (1)/(1) the proportion of 1 to 1, and by (1)/(2) the proportion of 1 to 2: if so, then it will follow, that if the proportions of 0 to 1 and of 1 to 1 be compounded by addition, the proportion arising will be the proportion of 1 to 2. But the proportion of 0 to 1 is infinitely little, that is, none. Therefore the proposition arising by composition will be that of 1 to 1, and equal (because of the symbol =) to the proportion of 1 to 2, and so 1 = 2. This also is so absurd that I dare say that you will not own it.

There may be another meaning yet: perhaps you mean that the uppermost quantity 0 + 1 is equal to the uppermost quantity 1; and the lowermost quantity 1 + 1 equal to the lowermost quantity 2: which is true. But how then in this equation (1)/(2) = (1)/(3) + (1)/(6)? Is the uppermost quantity 1 equal to the uppermost quantity 1 + 1; or the lowermost quantity 2 equal to the lowermost quantity 3 + 6? Therefore neither can this be your meaning. Unless you make your symbols more significant, you must not blame me for want of understanding them.

Let us now try what better success we shall have where the places are three, as here:

(0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (5)/(12) = (1)/(3) + (1)/(12):

If your symbols be fractions, the compound of them by addition is (5)/(4), for 0(1)/(4) and (4)/(4) make (5)/(4); and consequently (because of the symbol = ) (5)/(4) equal to (5)/(12), which is not to be allowed, and therefore that was not your meaning. If you meant that the proportions of 0 to 4 and of 1 to 4 and of 4 to 4 compounded, is equal to the proportion of 5 to 12, you will fall again into no less an inconvenience. For the proportion arising out of that composition will be the proportion of 1 to 4. For the proportion of 0 to 4 is infinitely little. Then to compound the other two, set them in this order 1. 4. 4. and you have a proportion compounded of 1 to 4 and of 4 to 4, namely, the proportion of the first to the last, which is of 1 to 4, which must be equal, by this your meaning, to the proportion of 5 to 12, and consequently as 5 to 12, so is 1 to 4, which you must not own. Lastly, if you mean that the uppermost quantities to the uppermost, and the lowermost to the lowermost in the first equation are equal, it is granted, but then again in the second equation it is false. It concerns your fame in the mathematics to look about how to justify these equations which are the premises to your conclusion following, namely, that the proportion arising is every where greater than sub-triple, or a third; and that the excess (that is, the excess above subtriple) perpetually decreaseth as the number of terms is augmented, as here (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. which I will show you plainly is false.

But first I wonder why you were so angry with me for saying you made proportion to consist in the quotient, as to tell me it was abominably false, and to justify it, cite your own words _penes quotientem_; do not you say here, the proportion is everywhere greater than subtriple, or (1)/(3)? And is not (1)/(3) the quotient of 1 divided by 3? You cannot say in this place that _penes_ is understood; for if it were expressed you would not be able to proceed.

But I return to your conclusion, that the excess of the proportion of the increasing quantities above the third part of so many times the greatest, decreaseth, as (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. For by this account in this row (0 + 1)/(1 + 1) = (1)/(2) where the quantity above exceeds the third part of the quantities below by (1)/(3), you make (1)/(3) equal to (1)/(6), which you do not mean. It may be said your meaning is, that the proportion of 1 to the subtriple of 2 which is (2)/(3), exceedeth what? I cannot imagine what, nor proceed further where the terms be but two. Let us therefore take the second row, that is, (0 + 1 + 4)/(4 + 4 + 4) = (5)/(12). The sum above is 5, the sum below is 12, the third part whereof is 4; if you mean, that the proportion of 5 to 4 exceeds the proportion of 4 to 12 (which is subtriple) by (1)/(12), you are out again. For 5 exceeds 4 by unity, which is (12)/(12). I do not think you will own such an equation as (12)/(12) = (1)/(12) Therefore I believe you mean (and your next proposition assures me of it), that the proportion of 5 to 4 exceeds subtriple proportion by the proportion of 1 to 12; if you do so, you are yet deceived.

For if the proportion of 5 to 4 exceeds subtriple proportion by the proportion of 1 to 12, then subtriple proportion, that is, of 4 to 12 added to the proportion of 1 to 12 must make the proportion of 5 to 4. But if you look on these quantities, 4, 12, 144, you will see, and must not dissemble, that the proportion of 4 to 12 is subtriple, and the proportion of 12 to 144 is the same with that of 1 to 12. Therefore by your assertion it must be as 5 to 4 so 4 to 144, which you must not own.

And yet this is manifestly your meaning, as appeareth in these words: “_Ut sit rationis provenientis excessus supra subtriplam ea quam habet unitas ad sextuplum numeri terminorum post 0, adeoque_,” which cannot be rendered in English, nor need to be. For you express yourself in the twentieth proposition very clearly; I noted it only that you may be more merciful hereafter to the stumblings of a hasty pen. For _excessus ea quam_ does not well, nor is to be well excused by _subauditur ratio_. Your twentieth proposition is this:

“_Si proponatur series quantitatum in duplicata ratione arithmetice proportionalium (sive juxta seriem numerorum quadraticorum) continue crescentium, a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem totidem maximæ æqualium subtriplam superabit; eritque excessus ea ratio quam habet unitas ad sextuplum numeri terminorum post 0, sive quam habet radix quadratica termini primi post 0 ad sextuplum radicis quadraticæ termini maximi._”

That is, if there be propounded a row of quantities in duplicate proportion of arithmetically-proportionals (or according to the row of square numbers) continually increasing, and beginning with a point or O. The proportion of that row to a row of so many equals to the greatest, shall be greater than subtriple proportion, and the excess shall be that proportion which unity hath to the sextuple of the number of terms after 0, or the same which the square root of the first number after 0, hath to the sextuple of the square root of the greatest.

For proof whereof you have no more here than _patet ex præcedentibus_; and no more before but _adeoque_. You do not well to pass over such curious propositions so slightly; none of the ancients did so, nor, that I remember, any man before yourself. The proposition is false, as you shall presently see.

Take, for example, any one of your rows: as (0 + 1 + 4)/(4 + 4 + 4). By this proportion of yours 1 + 4, which makes 5, is to 12 in more than subtriple proportion; by the proportion of 1 to the sextuple of 2 which is 12. Put in order these three quantities 5, 4, 12, and you must see the proportion of 5 to 12 is greater than the proportion of 4 to 12, that is, subtriple proportion, by the proportion of 5 to 4. But by your account the proportion of 5 to 4 is greater than that of 4 to 12 by the proportion of 1 to 12. Therefore, as 5 to 4 so is 1 to 12, which is a very strange paradox.

After this you bring in this consectary: “_Cum autem crescente numero terminorum excessus ille supra rationem subtriplam continue minuatur, ut tandem quovis assignabili minor evadat (ut patet) si in infinitum producatur, prorsus evaniturus est. Adeoque._”

That is, seeing as the number of terms increaseth, that excess above subtriple proportion continually decreaseth, so as at length it becomes less than any assignable (as is manifest) if it be produced infinitely, it shall utterly vanish, and so. And so what?

Sir, this consequence of yours is false. For two quantities being given, and the excess of the greater above the less, that excess may continually be decreased, and yet never quite vanish. Suppose any two unequal quantities differing by more than an unit, as 3 and 6, the excess 3, let 3 be diminished, first by an unit, and the excess will be 2, and the quantities will be 3 and 5; 5 is greater than 4, the excess 1. Again, let 1 be diminished and made (1)/(2), the excess 4 and the quantities 3 and 4(1)/(2), 4(1)/(2) is yet greater than 4. Again diminish the excess to (1)/(4), the quantities will be 3 and 4(1)/(4), yet still 4(1)/(4) is greater than 4. In the same manner you may proceed to (1)/(8) (1)/(16) (1)/(32), &c. infinitely; and yet you shall never come within an unit (though your unit stand for 100 miles) of the lesser quantity propounded 3, if that 3 stands for 300 miles. The excesses above subtriple proportion do not decrease in the manner you say it does, but in the manner which I now shall show you.

In the first row (0 + 1)/(1 + 1) a third of the quantities below is (2)/(3), set in order these three quantities 1 (2)/(9) (2)/(3). The first is 1, equal to the sum above, the last is (2)/(3), equal to the subtriple of the sum below. The middlemost is (2)/(9) subtriple to the last quantity (2)/(3). The excess of the proportion of 1 to (2)/(3) above the subtriple proportion of (2)/(9) to (2)/(3) is the proportion of 1 to (2)/(9) that is of 9 to 2, that is, of 18 to 4.

Secondly, in the second row, which is (0 + 1 + 4)/(4 + 4 + 4), a third of the sum below is 4, the sum above is 5. Set in order these quantities, 1, 5, 4, 12. There the proportion of 15 to 12 is the proportion of 5 to 4. The proportion of 4 to 12 is subtriple; the excess is the proportion of 15 to 4, which is less than the proportion of 18 to 4, as it ought to be; but not less by the proportion of (1)/(6) to (1)/(12) as you would have it.

Thirdly, in the third row, which is (0 + 1 + 4 + 9)/(9 + 9 + 9 + 9). A third of the sum below is 12, the sum above is 14. Set in order these quantities, 42, 4, 12. There the proportion of 42 to 12 is the same with that of 14 to 4. And the proportion of 4 to 12 subtriple, less than the former excess of 15 to 4. And so it goes on decreasing all the way in this manner, 18 to 4, 15 to 4, 14 to 4, &c. which differs very much from your 1 to 6, 1 to 12, 1 to 18, &c. and the cause of your mistake is this: you call the twelfth part of twelve (1)/(12), and the eighteenth part of thirty-six you call (1)/(18), and so of the rest. But what need of all those equations in symbols, to show that the proportion decreases; is there any man can doubt, but that the proportion of 1 to 2 is greater than that of 5 to 12, or that of 5 to 12 greater than that of 14 to 36, and so on continually forwards; or could you have fallen into this error, unless you had taken, as you have done in very many places of your _Elenchus_, the fractions (1)/(6) and (1)/(12), &c. which are the quotients of 1 divided by 6 and 12, for the very proportions of 1 to 6 and 1 to 12. But notwithstanding the excess of the proportions of the increasing quantities, to subtriple proportion decrease, still, as the number of terms increaseth, and that what proportions soever I shall assign, the decrement will in time (in time, I say, without proceeding _in infinitum_) produce a less, yet it does not follow that the row of increasing quantities shall ever be equal to the third part of the row of so many equals to the last or greatest. For it is not, I hope, a paradox to you, that in two rows of quantities the proportion of the excesses may decrease, and yet the excesses themselves increase, and do perpetually.

For in the second and third rows, which are (0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) and (0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) 5 exceeds the third part of 12 by a quarter of the square of 4, and 14 exceeds the third part of 36 by 2 quarters of the square of 4, and proceeding on, the sum of the increasing quantities where the terms are 5 (which sum is 30) exceedeth the third part of those below, (those below are 80, and their third part 26(2)/(3)) by 3 quarters and (1)/(2) a quarter of the square of 4, and when the terms are 6, the quantities above will exceed the third part of them below by 5 quarters of the square of 4. Would you have men believe, that the further they go, the excess of the increasing quantities above the third part of those below shall be so much the less? And yet the proportions of those above, to the thirds of those below, shall decrease eternally; and therefore your twenty-first proposition is false, namely this:

“_Si proponatur series infinita quantitatum in duplicata ratione arithmetice proportionalium (sive juxta seriem numerorum quadraticorum), continue crescentium a puncto sive 0 inchoatarum; erit illa ad seriem totidem maximæ æqualium, ut 1 ad 3._”

That is, if an infinite row of quantities be propounded in duplicate proportion of arithmetically-proportionals (or according to the row of quadratic numbers), continually increasing and beginning from a point or 0; that row shall be to the row of as many equals to the greatest, as 1 to 3. This is false, _ut patet ex præcedentibus_; and, consequently, all that you say in proof of the proportion of your _parabola_ to a _parallelogram_, or of the _spiral_ (the true _spiral_) to a _circle_ is in vain.

But your spiral puts me in mind of what you have under-written to the diagram of your proposition 5. _The spiral, in both figures, was to be continued whole to the middle, but, by the carelessness of the graver, it is in one figure_ manca, _in the other_ intercisa.

Truly, Sir, you will hardly make your reader believe that a graver could commit those faults without the help of your own copy, nor that it had been in your copy, if you had known how to describe a spiral line then as now. This I had not said, though truth, but that you are pleased to say, though not truth, that I attributed to the printer some faults of mine.

I come now to the thirty-ninth proposition, which is this:

“_Si proponatur series quantitatum in triplicata ratione arithmetice proportionalium (sive juxta seriem numerorum cubicorum), continue crescentium a puncto sive 0 inchoatarum (puta ut 0, 1, 8, 27, etc.), propositum sit inquirere quam habeat series illa rationem ad seriem totidem maximæ æqualium_:

“_Fiat investigatio per modum inductionis_ (_ut in prop. 1, et prop. 19_):

_Eritque_

(0 + 1 = 1)/(1 + 1 = 2) = (2)/(4) = (1)/(4) + (1)/(4)

(0 + 1 + 8 = 9)/(8 + 8 + 8 = 24) = (1)/(4) + (1)/(8)

(0 + 1 + 8 + 27 = 36)/(27 + 27 + 27 + 27 = 108) = (4)/(12)
= (1)/(4) + (1)/(12)

_Et sic deinceps._

“_Ratio proveniens est ubique major quam subquadrupla, sive (1)/(4). Excessus autem perpetuo decrescit, pro ut numerus terminorum augetur, puta (1)/(4) (1)/(8) (1)/(12) (1)/(16) etc. Aucto nimirum fractionis denominatore sive consequente rationis in singulis locis numero quaternatio, ut patet, ut sit rationis provenientis excessus supra subquadruplam ea quam habet unitas ad quadruplum numeri terminorum post 0 adeoque._”

That is, if a row of quantities be propounded in triplicate proportion of arithmetically proportionals (or according to the row of cubic numbers), continually increasing, and beginning from a point or 0, as 0, 1, 8, 27, 64, &c., let it be propounded to inquire, what proportion that row hath to a row of as many equals to the greatest.

Be it sought by way of induction, as in proposition 1 and 19.

The proposition arising is everywhere greater than subquadruple, or (1)/(4), and the excess perpetually decreaseth as the number of terms increaseth, as (1)/(4) (1)/(8) (1)/(12) (1)/(16) (1)/(20) &c. The denominator of the fraction, or consequent of the proportion, being in every place augmented by the number 4, as is manifest, so that the excess of the arising proportion above subquadruple is the same with that which an unit hath to the quadruple of the number of the terms after 0, and so. Here are just the same faults which are in proposition 19.

For, if (0)/(1) be a fraction, and (1)/(1) be a fraction, and (1)/(2) be another fraction, then this equation (0 + 1 = 1)/(1 + 1 = 2) is false. For this fraction (0)/(1) is equal to 0; and, therefore, we have (1)/(1) = (1)/(2), that is, the whole equal to half. But perhaps you do not mean them fractions, but proportions; and, consequently, that the proportion of 0 to 1, and of 1 to 1, compounded by addition (I say by addition, not that I, but that you think there is a composition of proportions by multiplication, which I shall show you anon is false), must be equal to the proportion of 1 to 2, which cannot be. For the proportion of 0 to 1 is infinitely little, that is, none at all; and, consequently, the proportion of 1 to 1 is equal to the proportion of 1 to 2, which is again absurd. There is no doubt but the whole number of 0 + 1 is equal to 1, and the whole number of 1 + 1 equal to 2. But, reckoning them as you do, not for whole numbers, but for fractions or proportions, the equations are false.

Again, your second equation, (2)/(4) = (1)/(4) + (1)/(4), though meant of fractions, that is, of quotients, it be true, and serve nothing to your purpose, yet, if it be meant of proportions, it is false. For the proportion of 1 to 4, and of 1 to 4 being compounded, are equal to the proportion of 1 to 16, and so you make the proportion of 2 to 4 equal to the proportion of 1 to 16, where, as it is but subquaduplicate, as you call it, or the quarter of it, as I call it. And, in the same manner, you may demonstrate to yourself the same fault in all the other rows of how many terms soever they consist. Therefore, you may give for lost this thirty-ninth proposition, as well as all the other thirty-eight that went before. As for the conclusion of it, which is, _that the excess of the arising proportion_, &c. They are the words of your fortieth proposition, where you express yourself better, and make your error more easy to be detected.

The proposition is this:

“_Si proponatur series quantitatum in triplicata ratione arithmetice proportionalium (sive juxta seriem numerorum cubicorum) continue crescentium a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem totidem maximæ æqualium subquadruplam superabit; eritque excessus ea ratio quam habet unitas ad quadruplum numeri terminorum post 0; sive quam habet radix cubica termini primi post 0 ad quadruplum radicis cubicæ termini maximi. Patet ex præcedente._

“_Quum autem crescente numero terminorum excessus ille supra rationem subquadruplam ita continuo minuatur, ut tandem quolibet assignabili minor evadat, ut patet, si in infinitum procedatur, prorsus evaniturus est, adeoque._

“_Patet ex propositione_ _præcedente._”

That is, if a row of quantities be propounded in triplicate proportion of arithmetically proportionals (or according to the row of cubic numbers), continually increasing, and beginning at a point or 0; the proportion which that row hath to a row of as many equals to the greatest, is greater than subquadruple proportion; and the excess is that proportion which one unit hath to the quadruple of the number of terms after 0; or, which the cubic root of the first term after 0 hath to the quadruple of the root of the greatest term.

It is manifest by the precedent propositions.

And, seeing the number of terms increasing, that excess above quadruple proportion doth so continually decrease, as that, at length, it becomes less than any proportion that can be assigned, as is manifest, if the proceeding be infinite, it shall quite vanish. And so

This conclusion was annexed to the end of your thirty-ninth proposition, as there proved. What cause you had to make a new proposition of it, without other proof than _patet ex præcedente_, I cannot imagine. But, howsoever, the proposition is false.

For example, set forth any of your rows, as this of fewer terms:

(0 + 1 + 8 + 27 = 36)/((27 + 27 + 27 + 27 =
108)

The row above is 36, the fourth part of the row below is 27. The quadruple of the number of terms after 0 is 12. Then, by your account, the proportion of 36 to 108 is greater than subquadruple proportion by the proportion of 1 to 12. For trial whereof, set in order these three quantities, 36, 27, 108. The proportion of 36 (the uppermost row) to 108 (the lowermost row) is compounded by addition of the proportions 36 to 27, and 27 to 108. And the proportion of 36 to 108, exceedeth the proportion of 27 to 108, by the proportion of 36 to 27. But the proportion of 27 to 108 is subquadruple proportion. Therefore, the proportion of 36 to 108 exceedeth subquadruple proportion, by the proportion of 36 to 27. And, by your account, by the proportion of 1 to 12; and, consequently, as 36 to 27, so is 1 to 12. Did you think such demonstrations as these should always pass?

Then, for your inference from the decrease of the proportions of the excess, to the vanishing of the excess itself, I have already showed it to be false; and by consequence that your next proposition, namely, the fortieth, is also false.

The proposition is this:

“_Si proponatur series infinita quantitatum in triplicata ratione arithmetice proportionalium (sive juxta seriem numerorum cubicorum), continue crescentium a puncto sive 0 inchoatarum, erit illa ad seriem totidem maximæ æqualium, ut 1 ad 4, patet ex præcedente._”

That is, if there be propounded an infinite row of quantities in triplicate proportion of arithmetically proportionals (or according to the row of cubic numbers), continually increasing, and beginning at a point or 0; it shall be to the row of as many equals to the greatest as 1 to 4. Manifest out of the precedent proposition.

Even as manifest as that 36, 27, 1, 12, are proportionals. Seeing, therefore, your doctrine of the spiral lines and the spaces is given by yourself for lost, and a vain attempt, your first forty-one propositions are undemonstrated, and the grounds of your demonstrations all false. The cause whereof is partly your taking quotient for proportion, and a point for 0, as you do in the first, sixteenth, and fortieth propositions, and in other places where you say, _beginning at a point or 0_, though now you deny you ever said either. There be very many places in your _Elenchus_, where you say both; and have no excuse for it, but that, in one of the places, you say the proportion is _penes quotientem_, which is to the same or no sense.

Your forty-second proposition is grounded on the fortieth; and therefore, though true, and demonstrated by others, is not demonstrated by you.

Your forty-third is this:

“_Pari methodo invenietur ratio seriei infinitæ quantitatum arithmetice proportionalium in ratione quadruplicata, quintuplicata, sextuplicata, etc., arithmetice proportionalium a puncto seu 0 inchoatarum, ad seriem totidem maximæ æqualium. Nempe in quadruplicata erit, ut 1 ad 5; in quintuplicata, ut 1 ad 6; in sextuplicata, ut 1 ad 7. Et sic deinceps._”

That is, by the same method will be found, the proportion of an infinite row of arithmetically proportionals, in proportion quadruplicate, quintuplicate, sextuplicate, &c., of arithmetically proportionals, beginning at a point or 0, to the row of as many equals to the greatest; namely, in quadruplicate, it shall be as 1 to 5; in quintuplicate, as 1 to 6; in sextuplicate, as 1 to 7; and so forth.

But by the same method that I have demonstrated, that the propositions 19, 20, 21, 39, 40, and 41, are false: any man else, that will examine the forty-third may find it false also. And, because all the rest of the propositions of your _Arithmetica Infinitorum_ depend on these, they may safely conclude, that there is nothing demonstrated in all that book, though it consist of 194 propositions. The proportions of your parabolocides to their parallelograms are true, but the demonstrations false, and infer the contrary. Nor were they ever demonstrated (at least the demonstrations are not extant) but by me; nor can they be demonstrated, but upon the same grounds, concerning the nature of proportion, which I have clearly laid, and you not understood. For, if you had, you could never have fallen into so gross an error as is this your book of _Arithmetica Infinitorum_, or that of the angle of contact. You may see by this, that your symbolic method is not only not at all inventive of new theorems, but also dangerous in expressing the old. If the best masters of symbolics think for all this you are in the right, let them declare it. I know how far the analysis by the powers of the lines extendeth, as well as the best of your half-learnt epistlers, that approve so easily of such analogisms as those, 5, 4, 1, 12, and 36, 27, 1, 12, &c.

It is well for you that they who have the disposing of the professors’ places take not upon them to be judges of geometry. For, if they did, seeing you confess you have read these doctrines in your school, you had been in danger of being put out of your place.

When the author of the paper wherein I am called Plagiary, and wherein the honour is taken from you of being the first inventor of these fine theorems, shall read this that I have here written, he will look to get no credit by it; especially if it be Roberval, which methinks it should not be. For he understands what proportion is, better than to make 5 to 4 the same with 1 to 12. Or to make, again, the proportion of 36 to 27 the same with that of 1 to 12; and innumerable _disproportionalites_ that may be inferred from the grounds you go on. But if it be Roberval indeed, that snatches this invention from you, when he shall see this burning coal hanging at it, he will let it fall again, for fear of spoiling his reputation.

But what shall I answer to the authority of the three great mathematicians that sent you those encomiastic letters. For the first, whom you say I use to praise, I shall take better heed hereafter of praising any man for his learning whilst he is young, further than that he is in a good way. But it seems he was in too ready a way of thinking very well of himself, as you do of yourself. For the muddiness of my brain I must confess it; but, Sir, ought not you to confess the same of yours? No, men of your tenets use not to do so. He wonders, say you, you thought it worth the while to foul your fingers about such a piece. It is well; every man abounds in his own sense. If you and I were to be compared by the compliments that are given us in private letters, both you and your complimentors would be out of countenance; which compliments, besides that which has been printed and published in the commendations of my writings, if it were put together, would make a greater volume than either of your libels. And truly, Sir, I had never answered your Elenchus as proceeding from Dr. Wallis, if I had not considered you also as the minister to execute the malice of that sort of people that are offended with my _Leviathan_.

As for the judgment of that public Professor that makes himself a witness of the goodness of your geometry, a man may easily see by the letter itself that he is a dunce. And for the English person of quality whom I know not, I can say no more yet than I can say of all three, that he is so ill a geometrician, as not to detect those gross paralogisms as infer that 5 to 4 and 1 to 12 are the same proportion. He came into the cry of those whom your title had deceived.

And now I shall let you see that the composition of proportion by multiplication, as it is in the fifth definition of the sixth element, is but another way of adding proportions one to another. Let the proportions be of 2 to 3, and of 4 to 5. Multiply 2 into 4 and 3 into 5, the proportion arising is of 8 to 15. Put in order these three quantities, 8, 12, 15. The proportion therefore of 8 to 15, compounded of the proportions of 8 to 12, (that is, of 2 to 3) and of 12 to 15, that is, of 4 to 5 by addition. Again, let the proportion be of 2 to 3, and of 4 to 5, multiply 2 into 5 and 3 into 4, the proportions arising is of 10 to 12. Put in order these three numbers, 10, 8, 12. The proportion 10 to 12 is compounded of the proportions of 10 to 8, that is of 5 to 4, and of 8 to 12, that is, of 2 to 3 by addition. I wonder you know not this.

I find not any more clamour against me for saying the proportion of 1 to 2 is double to that of 1 to 4.

Your book, you speak of, concerning proportion against _Meibomius_ is like to be very useful when neither of you both do understand what proportion is.

You take exceptions, as that I say, that _Euclid_ has but one word for _double_ and _duplicate_; which nevertheless was said very truly, and that word is sometimes διπλάσιος and sometimes διπλάσιων. And you think you have come off handsomely with asking me whether διπλάσιος and διπλασίων be one word.

Nor are you now of the mind you were, that a point is not _quantity unconsidered_, but that in an infinite series it may be safely neglected. What is _neglected_ but unconsidered.

Nor do you any more stand to it, that the _quotient_ is the _proportion_. And yet were these the main grounds of your _Elenchus_.

But you will say, perhaps, I do answer to the defence you have now made in this your _School Discipline_: ’tis true. But ’tis not because you answer never a word to my former objections against these propositions 19, 89; but because you do so shift and wriggle, and throw out ink, that I cannot perceive which way you go, nor need I, especially in your vindication of your _Arithmetica Infinitorum_. Only I must take notice that in the end of it, you have these words, “Well, _Arithmetica Infinitorum_ _is come off clear_” You see the contrary. For sprawling is no defence.

It is enough to me that I have clearly demonstrated both before sufficiently, and now again abundantly, that your book of _Arithmetica Infinitorum_ is all nought from the beginning to the end, and that thereby I have effected that your authority shall never hereafter be taken for a prejudice. And, therefore, they that have a desire to know the truth in the questions between us, will henceforth, if they be wise, examine my geometry, by attentive reading me in my own writings, and then examine, whether this writing of yours confute or enervate mine.

There is in my fifth lesson a proposition, with a diagram to it, to make good, I dare say, at least against you, my twentieth chapter concerning the dimension of a circle. If that demonstration be not shown to be false, your objections to that chapter, though by me rejected, come to nothing. I wonder why you pass it over in silence. But you are not, you say, bound to answer it. True, nor yet to defend what you have written against me.

Before I give over the examination of your geometry, I must tell you that your words, (p. 101 of your _School Discipline_), against the first corollary are untrue.

Your words are these: “_you affirm that the proportion of the parabola A B I to the parabola A F K is triplicate to the proportion of the time A B to A F, as it is in the English_.” This is not so. Let the reader turn to the place and judge. And going on you say, “_or of the impetus B I to F K as it is in the Latin_.” Nay, as it is in the English, and the other in the Latin. It is but your mistake; but a mistake is not easily excused in a false accusation.

Your exception to my saying, “_that the differences of two quantities is their proportion_,” (when they differ, as the no difference, when they be equal), might have been put in amongst other marks of your not sufficiently understanding the Latin tongue. _Differre_ and _differentia_ differ no more than _vivere_ and _vita_, which is nothing at all, but as the other words require that go with them, which other words you do not much use to consider. But _differre_ and _the quantity by which they differ_, are quite of another kind. _Differre_ (τὸ διαφέρειν, τὸ ὑπερέχειν) _differing_, _exceeding_, is not quantity, but relation. But the quantity by which they differ is always a certain and determined quantity, yet the word _differentia_ serves for both, and is to be understood by the coherence with that which went before. But I had said before, and expressly to prevent cavil, that relation is nothing but a comparison, and that proportion is nothing but relation of quantities, and so defined them, and therefore I did there use the word _differentia_ for _differing_, and not for the quantity which was left by subtraction. For a quantity is not a differing. This I thought the intelligent reader would of himself understand without putting me, instead of _differentia_, to use (as some do, and I shall never do) the mongrel word τὸ _differre_. And whereas in one only place for _differre ternario_ I have writ _ternarius_, if you had understood what was clearly expressed before, you might have been sure it was not my meaning, and therefore the excepting against it was either want of understanding, or want of candour, choose which you will.

You do not yet clear your doctrine of _condensation_ and _rarefaction_. But I believe you will by degrees become satisfied that they who say the same numerical body may be sometimes greater, sometimes less, speak absurdly, and that _condensation_ and _rarefaction_ here, and _definitive_ and _circumscriptive_, and some other of your distinctions elsewhere are but snares, such as school divines have invented

——ᾥσπερ άράχνης
Ὀυλόμενος χέζει ἀλύσεις μυίαις ἀθαρέσσι,

to entangle shallow wits.

And that that distinction which you bring here, “_that it is of the same quantity while it is in the same place, but it may be of a different quantity when it goes out of its place_,” (as if the place added to, or took any quantity from the body placed), is nothing but mere words. It is true that the body which swells changeth place, but it is not by becoming itself a greater body, but by admixtion of air or other body, as when water riseth up in boiling, it taketh in some parts of air. But seeing the first place of the body is to the body equal, and the second place equal to the same body, the places must also be equal to one another, and consequently the dimensions of the body remain equal in both places.

Sir, when I said that such doctrine was taught in the Universities, I did not speak against the Universities, but against such as you. I have done with your geometry, which is one στιγμὴ.

RURAL LANGUAGE.