Stigmai ageōmetrias, agroichias, antipoliteas, amatheias, or, Markes of the absurd geometry, rural language, Scottish church-politicks, and barbarismes of John Wallis professor of geometry and doctor of divinity by Thomas Hobbes.

TO DOCTOR WALLIS In answer to his SCHOOLE DISCIPLINE.

SIR,

WHen ••nprovok'd you addressed unto me in your Elenchus your ha••sh com∣plement with great security, wantonly to shew your wit, I confesse 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 calm∣ly I can argue with you about your Geometry, and other parts of Learning.

I shall in the first part confer with you about your Arit••metica Infinitorum, and after∣wards compare our manner of Elocution; then your Politicks; and last of all your Grammar and Criticks. Your spirall line is condemned by him whose Authority you use to prove me a Plagiary, (that is, a man that st••aleth other mens inventions, and arro∣gates them to himself) whether it be Roberval or not that w••it that paper, I am not certain. But I think I shall be shortly, but whosoever it be, his authority will serve no lesse to shew that your Doctrine of the sp••rall line from the fi••th to the eighteenth pro∣position 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 my self con∣cerning that name of Plagiary; And whereas he challenges the invention of your Me∣thod 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 a shamed to own it; and those that writ those Encomiastick Epistles to you ashamed of the Honour they meant to you: I passe therefore to the ninteenth proposition, which in L••tine is this.

Your Geometry.

Si proponatur series Quantitatum in duplicata ratione Arithmetice. proportionalium (sive juxta seriem numerorum Quadraticorum) continu•• crescentium, à puncto vel o inchoa••arum, (puta ut 0. 1. 4. 9. 16. &c.) propositum sit, inquirere quam habeat illa rat••onem ad seriem totid••m maximae aequalium.

Fiat Investigatio per modum inductioni•• ut (in prop. 1.)

Eritque

• ...〈 math 〉〈 math 〉
• ...〈 math 〉〈 math 〉
• 〈 math 〉〈 math 〉 & sic dein••eps.

Ratio proveniens est ubique major quam subtripla seu ⅓▪; Excessus autem perpetuo decresci•• prout▪ numerus terminorum augetur (puta 1/6 1/12 1/18 1/30 &c.) aucto nimirum fractionis deno∣minatore sive consequente rationis in singulis locis numero senario (ut pa••et) ut ••it rationis provenientis excessus supra subtriplam, Ea quam habet unitas ad sextuplum numeri terminorum posto; adeo{que}.

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 num∣bers) continually increasing; and beginning at a point or 0; let it be propounded to finde 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 every where greater then subtriple, or ⅓, And the excesse perpetually decreaseth as the number of termes is augmented, as here, 1/6 1/12 1/18 1/24 1/30 &c. the denominator of the fraction being in every place augmented by the number six (as is manifest) so that the excesse of the rising proportion above subtriple is the same which unity hath to six times the number of termes after 0; and so.

Sir, In these your Characters I understand by the crosse + 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 betwen which they are placed are one to another equall; This is your meaning, or you should have told us what you meant else: I un∣derstand 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 Symboles no further; but must confer with your self about the rest.

And first I ask you (because fractions are commonly written in that manner) whe∣ther in the uppermost row (which is 〈 math 〉〈 math 〉) 0/1 be a fraction, 1/1 be a fra∣ction,

½ be a fraction, that is to say, a part of an unite, and if you will (for the cy∣phers sake) whether 0/1, be an infinitely little part of 1; and whether 1/1, or 1 divided by 1 signifie an unity; if that be your meaning, then the fractio•• 0/1 added to the fra∣ction 1/1 is equal to the fraction ½: But the fraction 0/1 is equal to 0; therefore the fraction 0/1 + 1/1 is equal to the fraction 1/1; and 1/1 equal to ½ which you will▪ confesse 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 consequent∣ly by 1/1 the proportion of 1 to 1, and by ½ 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 ••e 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 proportion arising by composition will be that of 1, to 1, and equall (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 quantitie 0 + 1 is equal to the uppermost q••antity 1; and the lowermost quantity 1 + 1 equal to the lowermost quantity 2: Which is true: but how then in this equation ½ = ⅓ + ⅙ is the uppermost quantity 1 equal to the uppermost quantity 1 + 1; or the lower most quantity 〈◊〉〈◊〉 equal to the lowermost quantity 3 + 6? Therefore neither can this be your meaning; unlesse you make your symbols more significant, you must not blame me for want of understanding them.

Let us now try what better successe we s••all have where the places are three, as here 〈 math 〉〈 math 〉: If your Symbols be fractions, the compound of them by addition is 5/4; For 0, ¼ and 4/4 make 5/4; and consequently (because of the Sym∣bole =) 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 & of 1 to 4 & 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, tis granted, but then again in the second Equation it is false. It concerns your fame in the Mathematicks to look about how to justifie these Equations which are the premises to your conclusion following, namely, that the proportion arising is every where greater then subtriple, 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 shew 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 justifie it, cite your own words Penes Quotientem, do not you say here, the proportion is every where greater then subtriple, or ⅓? And is not ⅓ the quotient of 1. divided by 3? You cannot say in this place that Penes is understood; for if it were expressed you would ••ot be able to proceed.

But I return to your conclusion, that the ex••ess of the proportion of the increasing quantities above the third part of so many t••mes the greatest, decreaseth as 1/6 1/12 1/18 1/24 1/30 &c. For by this accompt in this row 〈 math 〉〈 math 〉 where the quantitie above exceeds the third part of the quantities below by ⅓' you make ⅓ equal to ⅙' whi••h you do not mean. It may be said your meaning is that the proportion of 〈◊〉〈◊〉. to the sub∣triple of 2. which is ⅔ exceedeth what? I cannot imagine what, nor proceed further where the ter••s be but two. Let us therefore take the second row, that is, 〈 math 〉〈 math 〉 The summe above is 5. the summe below is 12. the third part where∣of 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 unitie, which is 12/12 I do not think you will own such an equation as 〈 math 〉〈 math 〉. Therefore I believe you mean (and your next proposition assures me of it) that the proportion of 5. to 4. ex∣ceeds subtriple proportion by the proportion of 1. to 12. if you do so, you are yet de∣ceived.

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 ••ee 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 asser∣tion it must be as 5. to 4. so 4. to 144. which you must not own.

And yet this is manifestly your meaning as apppeareth in th••se words; u•• 〈◊〉〈◊〉 rationis pr••venien••is excessus supra subtriplam ••a quam ••abet unitas ad sextuplum numeri terminorum post 0, adeoque, which cannot be rendered in English, nor need to be. For you ex∣press your self in the 20th. pr••position 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 20th. proposition is this▪

Siprop••natur serie•• quantitatum in duplicata ratione Arithmeticè proportionalium (sive jux∣••a seriem Numerorum Quadraticorum) continuè crescentium, a puncto vel O in choatarum, ratio quam habet illa ad seriem to••idem maximae aequalium subtriplam superabit; eritque excessus ea ratio qua•• habet unitas ad sex••uplum n••meri terminorum post 0, sive quam habet radix Quadra∣ticae termini primi post 0 ad sextuplum radicis Quadraticae termini maximi.

That is, if there be propounded a row of quantities in duplicate proportion of A∣rithmetically-proportionals (or according to the row of square numbers) continually increasing, and beginnin gwith a point or 0. The proportion of that row to a row of so

many equals to the greatest, shall be greater then subtriple proportion, and the excesse s••all be that proportion which unity h••th to the sextuple of the number of termes after 0, or the same which the square roo•• 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, then pa••e•• ex praeceden••••bus; and no more before but adeo{que}. You do not we••l to passe over such curious propositions so slightly; none of the Antients did so; nor, that I remember, any man before your self. The pro∣position is false, as you shall presently see.

Take for example any one of your rows; as 〈 math 〉〈 math 〉. By this proportion of yours 1 + 4 which makes 5 is to 12 in more then 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 that the proportion of 5 to 12 is greater then the proportion of 4 to 12, that is, subtriple proportion, by the proprtion of 5 to 4. But by your account the propor∣tion of 5 to 4 is greater then that of 4 to 12 by the proportion of 1 to 12. Therefore as 5 to 4 so is 1 to 1••. which is a very strang Parodox.

After this you bring in this Consectary.

Cum autem cresente numero terminorum excessus ille supra rationem sub••riplam conninuò minu∣atur, ut tandem quovis assignabili minor eva••••t (un p••tet) si in in••ini••um producatur prorsus eva∣niturus est. Adeoque.

That is, seeing as the number of tearms encreaseth, that excesse above subtriple pro∣portion continually decre••seth, so as at length it be••omes lesse then any assignable (as is manifes••) 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 ex∣cesse of the greater above the lesse, that excesse may continually be decreased, and ye•• never quite vanish. Suppose any two unequal quantities differing by more then an unite, as 3 and 6, the excesse 3, let three be diminished, fi••st by an unite, and the ezcesse will be 2 and the quantities will be 3 and 5. 5 is greater then 4; the excesse 1. Again, let 1 be diminished and made ½. ••he excesse ½ and the quantities 3 and 4½. 4½ is yet grea∣ter then 4. Again diminish the excesse to ¼, the quantities will be 3 and 4¼ yet still 4¼ is greater then 4. In the same manner you may proceed to ⅛, 1/16, 1/32, &c. Infinitely; and yet you shall never come within an unite (though your unite stand for 100 mile) of the lesser quantity propounded 3, if that 3 stand for 300 Mile. The exce••ses above subtriple proportion do not d••creas•• in ••he manner you say it does, but in the manner which I shall now shew you.

In this first row 〈 math 〉〈 math 〉 a third of the quantities below is ⅔. set in order these thre3 quantities 1. 2/9. ⅔. The first is 1 equal to the sum above, the last is ⅔ equal to the subtriple of the sum below. The middlemost is 2/9 subtriple to the last quantity ⅔.

The excesse of the proportion of 1 to ⅔ above the subtriple proportion of2/9 to ⅔ 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 〈 math 〉〈 math 〉 a third of the sum below is 4 the sum above is 5. Set in order these quantities 15. 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 excesse is the proportion of 15 to 4, which is lesse then the proportion of 18 to 4. as it ought to be; b••t not lesse by the proportion of ⅙ to 1/12, as you would have it.

Thirdly, in the third row, which is 〈 math 〉〈 math 〉. A third of the sum below is 12, the su•• above is 14. Set in order these quantities 4 〈◊〉〈◊〉. 4. 12. There the proportion of 42 to 12. is the same, with that of 14 to 4. And the p••oportion of 4 to 12 subtriple, lesse then the former excesse of 15 to 4. And so it goes on deceasing 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 ••ix, you call 1/18, and so of the rest. But what need all those equations in Symbols, to shew that the proportion decreases; is there any man can doubt, but th••t the propartion of 1 to 2 is greater then that of 5 to 12, or that of 5 to 12 greater then that of 14 to 36, and so continually forwards; or could you have fallen into this errour, unlesse you had taken, as you have done in very many places of your Elenchus, the Fra▪∣ctions ⅙ 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 excesse of the proportions of the encreasing quanti••ies, to subtriple proportion decrease stil, as the number of tearms increaseth, and that what proportions soever I shall assigne, the decrement will in time (in time I say without proceeding in in••initum) produce a lesse, yet it does not follow that the row of the increasing qu••ntities shall ever be equall to the third part of the row of so many equalls 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 them selves encrease, and do perpetually.

For in the second and third rows, which are 〈 math 〉〈 math 〉 and 〈 math 〉〈 math 〉 5 exceeds the third part of 12, by a quarter of the square of fo••r, and 14 exceedes the third part of 36 by 2 quarters of the square of 4, and proceeding on, the sum of the in∣creasing quantities where the termes are 5 (which sum is 30) exceedeth the third part of those below (those below are 80 and their third part 26⅔ by 3 quarters and ½ a quarter of the square of 4. and when the tearms are 6 the quantities above will ex∣ceed the third part of them below by 5 quarte••s of the square of four. Would you have ••en beleeve, that the forther you go, the excesse of the increasing quantities above the third p••r•• of those below shall be so much the lesse? And yet the proportions of those above, to the the thirds of those below, shall decrease eternally; and therefore your 〈◊〉〈◊〉 proposit••on is ••alse, namely this.

Siproponatur series Infinita quantitatum in duplic••t•• ratione Arithmeticè proportionalium (sive juxta seriem numerorum quadraticorum) continué crescentium 〈◊〉〈◊〉 puncto ••ive O inchoata••um; eri•• illa ad seriem to••idem maximae aequalium, ut 1 ad 3.

That i••, if an infinite row of quantities be propounded in duplicate proportion of A∣rithmetically proportio••alls (or ••ccording to the row of quadratick numbers) continual∣ly increasing and beginning from a point or 0; that row shall be to the row of as ma∣ny equalls to the greatest, as 1 to 3. This is false, ut patet ex praecedentibus, and conse∣quently all that yo•• say in proof ••f the proportion of your Parabola to a parallelogram, or of the spiral (he true spiral) to a circle is in vain.

But your spiral puts me in mind of what you h••ve underwritten to the diagramme of your prop 5. The spirall in both f••gures was to be continued whole to the middle, but by the carelessnesse of the Graver it is in one figure manca, in the other intercis••.

T••uly Sir, you will hardly make your Reader beleeve that a Graver could ••ommit those faults without the help of your own Coppy, nor that it had been in your coppy, 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 f••ults of mine;

I come now to the thirty ninth proposition which is this

Si proponatur series quantitatum in triplicata ratione Arithmeticè proportionalium (sive juxta seriem numerorum cubicorum) continuè cresentium a puncto sive O inc••oata••um, (puta ut O. 1. 8. 27. &c.) propositum sit inquirere quam habeat series illa rationem ad seriem totidem maximae ••qualium.

Fiat investigatio per modum Inductionis (ut in prop. 1 & prop 19.)

Eritque

• ...〈 math 〉〈 math 〉
• ...〈 math 〉〈 math 〉
• 〈 math 〉〈 math 〉 Et sic d••inceps

Ratio proveniens est ubique major quam subquadrupla, sive ¼. Excessus autem perp••••uo decrescit, p••o ut numerus terminor••m augetur, puta ¼. ⅛. 1/12. 1/16 &c. Aucto nimirum fractionis denominator•• ••ive cansequente rationis in ••ingulis locis numero quaternatio (ut patet) ut sit ra∣ti••nis provenie•…•… ••xcess•…•… supra subquadruplam e•• quam habet unitas ad Quadruplum numeri terminorum post 0. Adeo{que}.

That is, if a row of quantities be propounded in triplicate proportion of Arithmetically proportionalls (or according the row of cubiqūe numbers) continually encreasing, and beginning from a point or o (as 0. 1. 8. 27. 64. &c.) let it be propounded to enquire what proportion that row hath to a row of as many equalls to the greatest••

Be it sought by way of induction, (as in prop. 1. 19).

The proportion arising is every where greater then subquadruple, or ¼, and the ex∣cesse perpetually decreaseth as the number of tearms increaseth as ¼, ⅛, 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 excesse of the arising proportion above subquadruple is the same with that which an unite hath to the quadruple of the number of the tearms after 0. And so. Here are just the same faults which are in prop. 19.

For if 0/1 be a fraction, and 1/1 be a fraction and ½ be another fraction, then this e∣quation 〈 math 〉〈 math 〉 is false. For this fraction 0/1 is equal, to 0▪ and therefore we have 1/1 = ½, that is the whole equal to half. But perhaps you do not mean them fracti∣ons, 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 shew you anon is f••lse) 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 e∣quall 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 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 = ¼ + ¼ 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 tearms soever they consist, Therefore you may give for lost this 39. prop. as well as all the other 38 that went before. As for the conclusion of it, which is, that the excesse of the arising proportion &c. They are the words of your 40▪ proposition, where you▪ e••presse your self better, and make your errour more easie to be detected.

The proposition is this.

Si proponatur series quantitatum in triplicata ratione Arithmeticè proportionalium (sive juxta seriem numerorum cubicorum) continue crescentium à puncto velo inchoa••arum, ratio quam habet illa ad seriem totidem maximae aqualiùm 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 cubicae te••mini maximi. Pat•••• ex praecedente.

Quum autem crescente numero terminorum excessus ille supra rationem subquadruplam i••a con∣tinuo

minuatur, ut tandem qu••libet assignabili minor evadat, (u•• pa••et) si in Infinitum proce∣datur, prorsus evaniturus est. Ade••que.

Paret ex prop. praecedent.

That is; If a row of quantlties be propounded in triplicate proportion of Arithmeti∣cally proportionalls (or according to the row of cubick numbers) continually en∣creasing; and beginning at a point or 0; The proportion which that row ••ath to a row of as many equals to the greatest is greater then subquadruple proportion; and the ex∣cesse is that proportion which one unite hath to the Quadruple of the number of termes after 0. Or which the cubick root of the first term after 0 hath to the quadruple of the root of the greatest tearm.

It is manifest by the precedent propositions.

And seeing, the number of tearmes increasing, that excesse above quadruple propor∣tion doth so continually decrease, as that at length it becomes lesse then 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 39 proposition; as there prooved. What cause you had to make a new proposition of it, without other proof then pate•• ex praecedente, I cannot imagine. But howsoever the proposition is false.

For example, set forth any of your rowes, as this of fower termes. 〈 math 〉〈 math 〉

The row above is 36▪ the fourth part of the row below is 27. The quadruple of the number of termes after 0 is 12▪ then by your accompt, the proportion of 36 to 108 is greater then subquadruple proportion by the proportion of 1 to 12. For trial where∣of 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 proportion•• 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 alwayes passe?

Then for your inference from the decrease of the proportions of the excesse to the vanishing of the excesse it self, I have already shewed it to be false; and by consequence that your next Proposition, namely, the 40 is also false.

The proposition is this.

Si pr••ponatur series infinita quantitatum in triplicata ratione Arithmeticé proportionalium (sive juxta seriem numerorum cubicorum) continuè crescentium à puncto sive 0 inchoatarum, erit illa ad seriem totidem maximae aequalium, ut 1 ad 4. pa••et ex praecedente.

That is, If there be propounded an infinite row of quantities in triplicate proportion of Arithmetically proportionalls (or according to the row of cubick numbers) continual∣ly increasing, and beginning at a point or 0; it shall be to the row of as many equalls to the greatest as 1 to 4. Manifest out of the precedent proposition.

Even as manifest as that 36. 27. 1. 12. Are proportionalls: seeing therefore your Do∣ctrine of the spiral lines and spaces is given by your self for lost, and a vaine attempt, your first 41 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 1. 16 and 40. propositions and in other places where you say be∣ginning at a point or 0; though now you denie you ever said either. There be very ma∣nay 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 p••nes quotientem, which is to the same or no sense.

Your 42 proportion is grounded on the 40; and therefore though true, and demon∣strated by others, is not demonstrated by you.

Your 43 is this.

Pari methodo invenietur ratio seriei infinitae quantitatum Arithmeticè proportionalium in rati∣one quadruplicata, quintuplicata, sextuplicata, etc Arithmeticè proportionalium à puncto seu 0 in choatarum, ad seriem totidem maximae aequalium. Nempe in quadruplicata ••••it, ut 1 ad 5; in quintuplicata ut 1 ad 6; in sextuplicata ut 1 ad 7.

Et sic d••inceps.

That is, By the same method will be found the proportion of an infinite row of Arith∣metically proportionalls in proportion quadruplicate, quintuplicate, sextuplicate &c. of Arithmetically proportionalls beginning at a point or 0, to the row of as many equalls to the greatest; Namely, in quadruplicate, it shall be as 1 to 5, in quintuplicate as 1 to 6; in sextuplicat•• as 1 to 7; and so forth.

But by the same method that I have demonstrated that the 19. 20. 21. 39. 40. 41. Pro∣positions are false, any man else that will examine the 43 may finde it false also. And be∣cause 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 Parabolo••ides to their Parallel∣logrammes 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 fal∣len into so grosse an errour as is this your Book of Arithmetica Infinitorum, or that of the Angle of Contact. You may see by this that your symbolick Method is not onely, not at all inventive of new Theormes, but also dangerous in expressing the old. If the best Masters of Symbolicks 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-learn't Epistlers, that approve so easily of such An••logismes 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 confesse you have

read these Doctrines in your School•• you had been in danger of being put out of your place.

When the Author of ••he paper wherein I am called Plagiary, and wherein the honor is taken from you of being the first inventor of these fine Theoremes, shall read this that I have here written, he will look to get no credit by it; especially if it be Roberval, which me thinkes it should not be. For he understands what proportion is, better then 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 Robeval 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 Mathemiticians that sent you those Encomiastick letters. For the first, whom you say I use to praise, I shall take better ••eed hereafter of praising any man for his Learning whilst he is young, further then that he is in a good way. But it seemes he was in too ready a way of thinking very well of himself, as you do of your self. For the muddinesse of my brain, I must confesse it. But ••r, Ought not you to confesse 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 ••o••••l your fingers about such a piece. Tis well; Every man abounds in his own sense. If you and I were to be compared by the complements that are given us in p••ivate letters, both you and your Complementors would be out of Countenance; which ••omplements, besides that which has been printed and published in the Commendations of my writings, if it were put together would make a greater volume then either of your Libels. And true∣ly Sir, I had never answered your Elenchus as proceeding from Dr. Walli••▪ if I had not considered you also as the Minister to execute the malice of that sort of people that are offended with my L••viathan.

As for the judgement of that Publick Professor that makes himself a witnesse of the goodnesse of your Geometry, a man may easily see by the letter it selfe that he is a dun••. And for the English person of quality whom I know not, I can say no more yet, then I can say of all three, that he is so ill a Geometrician, as not to detect those grosse Paralo∣gismes as in••er 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 5 d••f. of the 6 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 pro∣portion arising is of 8 to 15. Put in order these three quantities 8. 12. 15. The propor∣tion 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 & 4 to 5 multiply 2 into 5 and 3 into 4 the proportion arising is of 10 to 12. Put in order these 3 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 finde not any more clamour against me for saying the proportion of 1 to 2 is doublé to that of 1 to 4.

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

You take e••ceptions at that I say, that Eucilde has but one word for double and duplicate; which neverthelesse was said very truely, and that word is sometimes 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 and some∣times 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉▪ And you think you come of handsomly with asking me whether 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 and 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 be one word.

Nor are you now of the minde you were, that a point is not quantity unconsidered; but that in an infinite series it may be safely neglected. What is neglected but unconsi∣dered?

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 not 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 objection against these prop. 19▪ 39. 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. Onely I must take notice that in the end of it, you have these words, well, Arithmetica Infinitorum is come off clear. You see the con∣trary. For sprawling is no defence.

It is enough to me, that I have clearly demonstrated both, before sufficienly, and now again abundantly, that your Book of Arithmetica In••••ni••orum is all nought from the be∣ginning 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 Geome∣try by attentive reading me in my own writings, and then examine, whether this writing of yours consute or enervate mine.

There is in my 5 lesson a proposition, with a diagramme to it to make good, (I dare say,) at least against you, my 20. Chapter concerning the dimension of a Circle. If that demonstration be not shewn to be false, your objections to that Chapter (though by me rejected) come to nothing. I wonder why you passe it over in silence. But you are not, you say, bound to answer it. True, nor yet to defend what your have written against me.

Before I give over the examination of your Geometry, I must tell you that your words (pag. 〈◊〉〈◊〉 of this your Schoole Discipline) again the first Corollary are untrue.

Your words are these, you aff•…•…n 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 imp•…•… B I to F K, as it is in the Latine. Nay, as it is in the English, and the other in the Latine. Tis 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 a∣mongst other marks of your not sufficiently understanding the Latine tongue. Differre and Differentia differ no more then 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 con∣sider. But differre and the quantity by which they differ, are quite of another kinde. Di••∣ferre (〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉) differing, exceeding, is not quantity, but relation. But the quantity by which they differ is alwayes a certain and determined quantity, yet the word differrentia 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 le••t by substraction. For a quantity is not a differing. This I thought the intelligent Reader would of himself understand without putting me, in∣stead of differentia to use (as ••ome do, and I shall never do) the mongrell word 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 differre. And whereas in one onely place for differre ternario I have writ ternarius, If you had un∣derstood what was clearly exprest 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 Candou••; chuse which you will.

You do not yet clear your Doctrine of Condensation and Rarefa••lion. But I beleev•• you will be degrees become satisfied that they who say the same Numerical Body may ••e sometimes greater, sometimes lesse, speak absurdly, and that Condensation and Rarefacti∣on here, and definitive and ci••cumscriptive and some other of your distinctions elsewhere are but ••nares, such as School-Divines have invented

—〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉

to intangle 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 mee•• words. Tis true that the Body which swells changeth place; but it is not by becoming it self a greater body, but by admixtion of Aire or other body; as when water riseth up in boyling, it taketh in some parts of Aire. 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 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.