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.

362TO 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 363of 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.

364TO

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 365show 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 366subtripla 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, 367and 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 368to 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 01 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 369equation 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 370sum 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 371seriem 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. 372But 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 41 2, 41 2 is yet greater than 4. Again diminish the excess to 1 4, the quantities will be 3 and 41 4, yet still 41 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 373in 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 374any 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 262 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, 375that 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, 376though 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 377inquire, 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 378whole 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 379continuo minuatur, ut tandem quolibet assignabili minor evadat, ut patet, si in infinitum procedatur, prorsus evaniturus est, adeoque.

“Patet ex propositione præcedente.”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 soAnd 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, 380the 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.

381Even 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, 382it 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 383school, 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, 384besides 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.

385I 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 διπλασίωνδιπλασίων 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 386abundantly, 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 387of 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 388degrees 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.

As for your eloquence, let the reader judge whether yours or mine be the more muddy, though I in plain scolding should have outdone you, yet 389I have this excuse which you have not, that I did but answer your challenge at that weapon which you thought fit to choose. The catalogue of the hard language which you put in at pages 3 and 4 of your School Discipline, I acknowledge to be mine, and would have been content you had put in all. The titles you say I give you of fools, beasts, and asses, I do not give you, but drive back upon you, which is no more than not to own them; for the rest of the catalogue, I like it so well as you could not have pleased me better than by setting those passages together to make them more conspicuous; that is all the defence I will make to your accusations of that kind.

And now I would have you to consider whether you will make the like defence against the faults that I shall find in the language of your School Discipline.

I observe, first, the facetiousness of your title-page, “Due correction for Mr. Hobbes, or School Discipline, for not saying his Lessons right.” What a quibble is this upon the word lesson; besides, you know it has taken wind; for you vented it amongst your acquaintance at Oxford then when my Lessons were but upon the press. Do you think if you had pretermitted that piece of wit, the opinion of your judgment would have been ere the less? But you were not content with this, but must make this metaphor from the rod to take up a considerable part of your book, in which there is scarce anything that yourself can think wittily said besides it. Consider also these words of yours: “It is to be hoped that in time you may come to learn the language, for you be come to great A already.” And presently after, 390“were I great A, before I would be willing to be so used, I should wish myself little a a hundred times.” Sir, you are a doctor of divinity and a professor of geometry, but do not deceive yourself, this does not pass for wit in these parts, no, nor generally at Oxford; I have acquaintance there that will blush at the reading it.

Again, in another place you have these words: “Then you catechize us, ‘what is your name? Are you geometricians? Who gave you that name,’” &c. Besides in other places such abundance of the like insipid conceits, as would make men think, if they were no otherwise acquainted with the University but by reading your books, that the dearth there of salt were very great. If you have any passage more like to salt than these are (excepting now and anon) you may do well to show it to your acquaintance, lest they despise you; for, since the detection of your geometry, you have nothing left you else to defend you from contempt. But I pass over this kind of eloquence, and come to somewhat yet more rural.

Page 27, line 1, you say I have given Euclid his lurry. And again, page 129, line 11, “and now he is left to learn his lurry.” I understand not the word lurry. I never read it before, nor heard it, as I remember, but once, and that was when a clown threatening another clown said he would give him such a lurry come poop, &c. Such words as these do not become a learned mouth, much less are fit to be registered in the public writings of a doctor of divinity. In another place you have these words, “just the same to a cow’s thumb,” a pretty adage.

391Page 2, “But prithee tell me.” And again, page 95, “prithee tell me, why dost thou ask me such a question,” and the like in many other places.

You cannot but know how easy it is and was for me to have spoken to you in the same language. Why did I not? Because I thought that amongst men that were civilly bred it would have redounded to my shame, as you have cause to fear that this will redound to yours. But what moved you to speak in that manner? Were you angry? If I thought that the cause, I could pardon it the sooner, but it must be very great anger that can put a man, that professeth to teach good manners, so much out of his wits as to fall into such a language as this of yours. It was perhaps an imagination that you were talking to your inferior, which I will not grant you, nor will the heralds, I believe, trouble themselves to decide the question. But, howsoever, I do not find that civil men use to speak so to their inferiors. If you grant my learning but to be equal to yours, (which you may certainly do without very much disparaging of yourself abroad in the world), you may think it less insolence in me to speak so to you in respect of my age, than for you to speak so to me in respect of your young doctorship. You will find that for all your doctorship, your elders, if otherwise of as good repute as you, will be respected before you. But I am not sure that this language of yours proceeded from that cause; I am rather inclined to think you have not been enough in good company, and that there is still somewhat left in your manners for which the honest youths of Hedington and Hincsey may compare with you for good language, as great a doctor as you are.

392For my verses of the Peak, though they be as ill in my opinion as I believe they are in yours, and made long since, yet they are not so obscene as that they ought to be blamed by Dr. Wallis. I pray you, sir, whereas you have these words in your School Discipline, page 96, “unless you will say that one and the same motion may be now and anon too.” What was the reason you put these words, now and anon too, in a different character, that makes them to be more taken notice of? Do you think that the story of the minister that uttered his affection (if it be not a slander) not unlawfully but unseasonably, is not known to others as well as to you? What needed you then, when there was nothing that I had said could give the occasion, to use those words; there is nothing in my verses that do olere hircum so much as this of yours. I know what good you can receive by ruminating on such ideas, or cherishing of such thoughts. But I go on to other words of mine by you reproached, “you may as well seek the focus of the parabola of Dives and Lazarus,” which you say is mocking the Scripture; to which I answer only, that I intended not to mock the Scripture, but you, and that which was not meant for mocking was none. And thus you have a second στιγμὴ.

GRAMMAR AND CRITIQUES.

I come now to the comparison of our Grammar and Critiques. You object first against the signification I give of στιγμὴ, and say thus: “What should come into your cap (that, if you mark it, in a man that wears a square cap to one that wears a hat, is very witty) to make you think that στιγμὴ signifies 393a mark or brand with a hot iron? I perceive where the business lies, it was στίγμα run in your mind when you talked of στιγμὴ; and because the words are somewhat alike you jumble them both together.” Sir, I told you once before, you presume too much upon your first cogitations. Aristophanes, in Ranis, Act. V. Scen. 5,

Κἄν μὴ ταχέως ἥκωσι

Νὴ τεν Ἀπόλλο στίξας ἀυτοὺς.

The old commentator upon the word στίξας saith thus, ϛίξας ἀντὶ τοῦ ϛιγματίσας, ἠν γάρ ξένος. That is, στίξας for ϛιγματίσας, for he (Adimantus) was not a citizen. I hope the commentator does not here mock Aristophanes for jumbling ϛίξας and ϛιγματίσας together, for want of understanding Greek. No, ϛίξας and στιγματίσας signify the same, save that for branding I seldom read ϛιγματίσας but ϛίξας. For ϛίγμα does no more signify a brand with a hot iron, than ϛιγμὴ a point made also with a hot iron. They have both one common theme ϛίζω, which does not signify pungo, nor interpungo, nor inuro, for all your Lexicon, but notam imprimere, or pungendo notare, without any restriction to burning or punching. It is therefore no less proper to say that ϛιγμὴ is a mark with a hot iron, than to say the same of στίγμα. The difference is only this, that when they marked a slave, or a rascal, as you are not ignorant is usually done here at the assizes in the hand or shoulder with a hot iron, they called that ϛίγμα, not for the burning, but for the mark. And as it would have been called ϛίγμα that was imprinted on a slave, though made by staining or incision, so it is ϛιγμὴ, though done with a hot iron. And therefore there was no jumbling of those two words together, as for want of reading Greek authors, and 394by trusting too much to your dictionaries, which you say are proofs good enough for such a business, you were made to imagine. The use I have made thereof was to show that a point, both by the word ΣημεῖονΣημεῖον in Euclid, and by the word στιγμὴ in some others, was not nothing, but a visible mark, the ignorance whereof hath thrown you into so many paralogisms in geometry.

But do you think you can defend your Adducis Malleum as well as I have now defended my ϛιγμὴ? You have brought, I confess, above a hundred places of authors, where there is the word duco, or some of its compounds, but none of them will justify Adducis Malleum, and, excepting two of those places, you yourself seem to condemn them all, comparing yours with none of the rest but with these two only, both out of Plautus, by you not well understood. The first is in Casina, Act. V. Scen. 2, “Ubi intro hanc novam nuptam deduxi, via recta, clavem abduxi;” which you, presently presuming of your first thoughts, a peculiar fault to men of your principles, assure yourself is right. But if you look on the place as Scaliger reads it, cited by the commentator, you will find it should be obduxi, and that clavis is there used for the bolt of the lock. Besides, he bolted it within. Whither then could he carry away the key? The place is to be rendered thus, when I had brought in this new bride I presently locked the door, and is this as bad every whit as Adducis Malleum? The second place is in Amphytruo, Act. I. Scen. 1, “Eam (cirneam), ut a matre fuerat natum, plenam vini eduxi meri,” which you interpret I brought out a flagon of wine, unlearnedly. They are 395the words of Mercury transformed into Sosia. And to try whether Mercury were Sosia or not, Sosia asked him where he was and what he did during the battle; to which Mercury answered, who knew where Sosia then was and what he did, I was in the cellar, where I filled a cirnea, and brought it up full of wine, pure as it came from its mother. By the mother of the wine meaning the vine, and alluding to the education of children, for ebibi said eduxi, and with an emphasis in meri, because cirnea (from Κφνάω, misceo) was a vessel wherein they put water to temper to their wine. Intimating that though the vessel was cirnea, yet the wine was merum. This is the true sense of the place; but you will have eduxi to be, I brought out, though he came not out himself. You see, sir, that neither this is so bad as Adducis Malleum.

But suppose out of some one place in some one blind author you had paralleled your Adducis Malleum, do you think it must therefore presently be held for good Latin? Why more than learn his lurry must be therefore thought good English a thousand years hence, because it will be read in Dr. Wallis’s long-lived works. But how do you construe this passage (1 Tim. ii. 15) of the Greek Testament: Σωθήσεται δὲ διὰ τῆς τεκνογονίας, ἐὰν μείνωσιν ἐν πίστει? You construe it thus: she shall be saved notwithstanding child-bearing, if (the woman) remain in the faith. Is child-bearing any obstacle to the salvation of women? You might as well have translated the first verse of the fifth of Romans in this manner, Being then justified by faith, we have peace with God notwithstanding our Lord Jesus Christ. I let pass your not finding in τεκνογονίας, as good a 396grammarian as you are, a nominative case to μείνωσιν. If you had remembered the place, 1 Pet. iii. 20, εσώθησαν δὶ ὑδατος, that is, they were saved in the waters, you would have thought your construction justified then very well; but you had been deceived, for διὰ does not there signify causam, ablationem impedimenti, but transitum; not cause or removing an impediment, but passage. Being come thus far, I found a friend that hath eased me of this dispute; for he showed me a letter written to himself from a learned man, that hath out of very good authors collected enough to decide all the grammatical questions between you and me, both Greek and Latin. He would not let me know his name, nor anything of him but only this, that he had better ornaments than to be willing to go clad abroad in the habit of a grammarian. But he gave me leave to make use of so much of the letter as I thought fit in this dispute, which I have done, and have added it to the end of this writing. But before I come to that, you must not take it ill, though I have done with your School Discipline, if I examine a little some other of your printed writings as you have examined mine; for neither you in geometry, nor such as you in church politics, cannot expect to publish any unwholesome doctrine without some antidotes from me, as long as I can hold a pen. But why did you answer nothing to my sixth Lesson? Because, you say, it concerned your colleague only. No, sir, it concerned you also, and chiefly, for I have not heard that your colleague holdeth those dangerous principles which I take notice of in you, in my sixth Lesson, page 350, upon the occasion of these 397words, not his but yours: “Perhaps you take the whole history of the fall of Adam for a fable, which is no wonder, seeing you say the rules of honouring and worshipping of God are to be taken from the laws.” In answer to which I said thus: “You that take so heinously, that I would have the rule of God’s worship in a Christian commonwealth to be taken from the laws, tell me from whom you would have them taken? From yourself? Why so, more than from me? From the bishops? Right, if the supreme power of the commonwealth will have it so; if not, why from them rather than from me? From a consistory of presbyters themselves, or joined with lay elders, whom they may sway as they please? Good, if the supreme governor of the commonwealth will have it so. If not, why from them rather than from me, or from any man else? They are wiser and learneder than I; it may be so, but it has not yet appeared. Howsoever, let that be granted. Is there any man so very a fool as to subject himself to the rules of other men in those things which do so nearly concern himself, for the title they assume of being wise and learned, unless they also have the sword which must protect them? But it seems you understand the sword as comprehended. If so, do not you then receive the rules of God’s worship from the civil power? Yes, doubtless; and you would expect, if your consistory had that sword, that no man should dare to exercise or teach any rules concerning God’s worship which were not by you allowed.”

This will be thought strong arguing, if you do not answer it. But the truth is, you could say 398nothing against it without too plainly discovering your disaffection to the government. And yet you have discovered it pretty well in your second Thesis, maintained in the Act at Oxford, 1654, and since by yourself published. This Thesis I shall speak briefly to.

SCOTCH CHURCH POLITICS.

You define ministers of the Gospel to be those to whom the preaching of the Gospel by their office is enjoined by Christ. Pray you, first, what do you mean by saying preaching ex officio is enjoined by Christ? Are they preachers ex officio, and afterwards enjoined to preach? Ex officio adds nothing to the definition; but a man may easily see your purpose to disjoin yourself from the state by inserting it.

Secondly, I desire to know in what manner you will be able out of this definition to prove yourself a minister? Did Christ himself immediately enjoin you to preach, or give you orders? No. Who then, some bishop, or minister, or ministers? Yes; by what authority? Are you sure they had authority immediately from Christ? No. How then are you sure but that they might have none? At least, some of them through whom your authority is derived might have none. And therefore if you run back for your authority towards the Apostles’ times but a matter of sixscore years, you will find your authority derived from the Pope, which words have a sound very unlike to the voice of the laws of England. And yet the Pope will not own you. There is no man doubts but that 399you hold that your office comes to you by successive imposition of hands from the time of the Apostles; which opinion in those gentle terms passeth well enough; but to say you derive your authority from thence, not through the authority of the sovereign power civil, is too rude to be endured in a state that would live in peace. In a word, you can never prove you are a minister, but by the supreme authority of the commonwealth. Why then do you not put some such clause into your definition? As thus, ministers of the Gospel are those to whom the preaching of the Gospel is enjoined by the sovereign power in the name of Christ. What harm is there in this definition, saving only it crosses the ambition of many men that hold your principles? Then you define the power of a minister thus: “The power of a minister is that which belongeth to a minister of the Gospel in virtue of the office he holds, inasmuch as he holds a public station, and is distinguished from private Christians. Such as is the power of preaching the Gospel, administering the sacrament, the use of ecclesiastical censures, and ordaining of ministers,” &c.

Again, how will you prove out of this definition that you, or any man else, hath the power of a minister, if it be not given him by him that is the sovereign of the commonwealth? For seeing, as I have now proved, it is from him that you must derive your ministry, you can have no other power than that which is limited in your orders, nor that neither longer than he thinks fit. For if he give it you for the instruction of his subjects in their duty, he may take it from you again whensoever he shall see you instruct them with undutiful 400and seditious principles. And if the sovereign power give me command, though without the ceremony of imposition of hands, to teach the doctrine of my Leviathan in the pulpit, why am not I, if my doctrine and life be as good as yours, a minister as well as you, and as public a person as you are? For public person, primarily, is none but the civil sovereign, and so secondarily, all that are employed in the execution of any part of the public charge. For all are his ministers, and therefore also Christ’s ministers because he is so; and other ministers are but his vicars, and ought not to do or say anything to his people contrary to the intention of the sovereign in giving them their commission.

Again, if you have in your commission a power to excommunicate, how can you think that your sovereign who gave you that commission, intended it for a commission to excommunicate himself? that is, as long as he stand excommunicate, to deprive him of his kingdom. If all subjects were of your mind, as I hope they will never be, they will have a very unquiet life. And yet this has, as I have often heard, been practised in Scotland, when ministers holding your principles had power enough, though no right, to do it.

And for administration of the sacraments, if by the supreme power of the commonwealth it were committed to such of the laity as know how it ought to be done as well as you, they would ipso facto be ministers as good as you. Likewise the right of ordination of ministers depends not now on the imposition of hands of a minister or presbytery, but on the authority of the Christian sovereign, Christ’s immediate vicar and supreme 401governor of all persons and judge of all causes, both spiritual and temporal, in his own dominions, which I believe you will not deny.

This being evident, what acts are those of yours which you call authoritative, and receive not from the authority of the civil power? A constable does the acts of a constable authoritatively in that sense. Therefore you can no otherwise claim your power than a constable claimeth his, who does not exercise his office in the constabulary of another. But you forget that the Scribes and the Pharisees sit no more in Moses’ chair.

You would have every minister to be a minister of the universal Church, and that it be lawful for you to preach your doctrine at Rome; if you would be pleased to try, you would find the contrary. You bring no argument for it that looks like reason. Examples prove nothing, where persons, times, and other circumstances differ; as they differ very much now when kings are Christians, from what they were then when kings persecuted Christians. It is easy to perceive what you aim at.

You would fain have market-day lectures set up by authority, (not by the authority of the civil power, but by the authority of example of the Apostles in the emission of preachers to the infidels), not knowing that any Christian may lawfully preach to the infidels; that is to say, proclaim unto them that Jesus is the Messiah, without need of being otherways made a minister, as the deacons did in the Apostles’ time; nor that many teachers, unless they can agree better, do anything else but prepare men for faction, nay, rather you know it well enough, but it conduces to your end 402upon the market-days to dispose at once both town and country, under a false pretence of obedience to God, to a neglecting of the commandments of the civil sovereign, and make the subject to be wholly ruled by yourselves, wherein you have already found yourselves deceived. You know how to trouble and sometimes undo a slack government, and had need to be warily looked to, but are not fit to hold the reins. And how should you, being men of so little judgment as not to see the necessity of unity in the governor, and of absolute obedience in the governed, as is manifest out of the place of your Elenchus above recited. The doctrine of the duty of private men in a commonwealth is much more difficult, not only than the knowledge of your symbols, but also than the knowledge of geometry itself. How then do you think, when you err so grossly in a few equations, and in the use of most common words, you should be fit to govern so great nations as England, Ireland, and Scotland, or so much as to teach them? For it is not reading but judgment that enables one man to teach another.

I have one thing more to add, and that is the disaffection I am charged withal to the universities. Concerning the Universities of Oxford and Cambridge, I ever held them for the greatest and noblest means of advancing learning of all kinds, where they should be therein employed, as being furnished with large endowments and other helps of study, and frequented with abundance of young gentlemen of good families and good breeding from their childhood. On the other side, in case the same means and the same wits should be employed in the advancing of the doctrines that tend 403to the weakening of the public, and strengthening of the power of any private ambitious party, they would also be very effectual for that; and consequently that if any doctrine tending to the diminishing of the civil power were taught there, not that the Universities were to blame, but only those men that in the universities, either in lectures, sermons, printed books, or theses, did teach such doctrine to their hearers or readers. Now you know very well that in the time of the Roman religion, the power of the Pope in England was upheld principally by such teachers in the universities. You know also how much the divines that held the same principles in Church government with you, have contributed to our late troubles. Can I therefore be justly taxed with disaffection to the universities for wishing this to be reformed? And it hath pleased God of late to reform it in a great measure, and indeed as I thought totally, when out comes this your Thesis boldly maintained to show the contrary. Nor can I yet call this your doctrine the doctrine of the university; but surely it will not be unreasonable to think so, if by public act of the university it be not disavowed, which done, and that as often as there shall be need, there can be no longer doubt but that the universities of England are not only the noblest of all Christian universities, but also absolutely, and of the greatest benefit to this commonwealth that can be imagined, except that benefit of the head itself that uniteth and ruleth all. I have not here particularized at length all the ill consequences that may be deduced from this Thesis of yours, because I may, when further provoked, have somewhat to say that is new. So much for the third ϛιγμὴ.

404

AN EXTRACT OF A LETTER CONCERNING THE GRAMMATICAL PART OF THE CONTROVERSY BETWEEN MR. HOBBES AND DR. WALLIS.

Mr. Hobbes hath these words: “Longitudinem percursam motu uniformi, cum impetu ubique ipsi B D æquali.” Dr. Wallis saith cum were better out, unless you would have impetus to be only a companion, not a cause. Mr. Hobbes answered it was the ablative case of the manner. The truth is the ablative case of the manner and cause both, may be used with the conjunction cum, as may be justified. Cicero in Lib. II. De Nat. Deorum: “Moliri aliquid cum labore operoso ac molesto;” and in his oration for Cæcina: “De se autem hoc prædicat, Antiocho Ebulii servo imperasse ut in Cæcinam advenientem cum ferro invaderet.” Let us see then what Dr. Wallis objects against Tully, where a casualty is imported, though we may use with in English, yet not cum in Latin; to kill with a sword, importing this to have an instrumental or causal influence, and not only that it hangs by the man’s side whilst some other weapon is made use of, is not in Latin occidere cum gladio, but gladio occidere. This shows that the Doctor hath not forgot his grammar, for the subsequent examples as well as this rule are borrowed thence. But yet he might have known that great personages have never confined themselves to this pedantry, but have chosen to walk in a greater latitude. Most of the elegancies and idioms of every language are exceptions to his grammar. But since Mr. Hobbes saith it is the ablative case of 405the manner, there is no doubt it may be expressed with cum. The Doctor in the meantime knew no more than what Lilly had taught him; Alvarez would have taught him more; and Vossius in his book, De Constructione, cap. XLVII. expressly teacheth, “Ablativos causæ, instrumenti, vel modi, non a verbo regi sed a præpositione omissa, a vel ab, de, e vel ex, præ, aut cum, ac præpositiones eas quandoque exprimi nisi quod cum ablativis instrumenti haud temere invenias;” and afterwards he saith, “non timere imitandum.” If this be so, then did Mr. Hobbes speak grammatically, and with Tully, but not usually. And might not one retort upon the Doctor, that Vossius is as great a critic as he?

His next reflection is upon prætendit scire, this he saith is an Anglicism. If this be all his accusation, upon this score we shall lose many expressions that are used by the best authors, which I take to be good Latinisms, though they be also Anglicisms, the latter being but an imitation of the former. The Doctor therefore was too fierce to condemn upon so general an account, that which was not to have been censured for being an Anglicism, unless also it had been no Latinism. Mr. Hobbes replies, that the printer had omitted se. He saith, this mends the matter a little. It is very likely, for then it is just such another Anglicism as that of Quintilian: “Cum loricatus in foro ambularet, prætendebat se id metu facere.” The Doctor certainly was very negligent, or else he could not have missed this in Robert Stephen. Or haply he was resolved to condemn Quintilian for this and that other Anglicism, “Ignorantia prætendi non potest,” as all those that have used 406prætendo, which are many and as good authors as Dr. Wallis, that makes his own encomiasts (not an Englishman amongst them) to write Anglicisms.

Then he blames “Tractatus hujus partis tertiæ, in qua motus et magnitudo per se et abstracte consideravimus, terminum hic statuo.” Here I must confess the exception is colourable, yet I can parallel it with the like objection made by Erasmus against Tully, out of whom Erasmus quotes this passage: “Diutius commorans Athenis, quoniam venti negabant solvendi facultatem, erat animus ad te scribere;” and excuses it thus, that Tully might have had at first in his thoughts volebam or statuebam, which he afterwards relinquished for erat animus, and did not remember what he had antecedently written, which did not vary from his succeeding thoughts, but words. And this excuse may pass with any who know that Mr. Hobbes values not the study of words, but as it serves to express his thoughts, which were the same whether he wrote in qua motus et magnitudo per se at abstracte considerati sunt or consideravimus. And if the Doctor will make this so capital, he must prove it voluntary, and show that it is greater than what is legible in the puny letter of his encomiast, whom he would have to be beyond exception.

Now follows his ridiculous apology for adducis malleum, ut occidas muscam. The cause why he did use that proverb, of his own phrasing, was this. Mr. Hobbes had taken a great deal of pains to demonstrate what Dr. Wallis thought he could have proved in short; upon this occasion he objects, adducis malleum ut occidas muscam, which I shall suppose he intended to English thus, you bring 407a beetle to kill a fly. Mr. Hobbes retorted, that adduco was not used in that sense. The Doctor vindicates himself thus: duco, deduco, reduco, perduco, produco, &c. signify the same thing, ergo, adduco may be used in that sense; which is a most ridiculous kind of arguing, where we are but to take up our language from others, and not to coin new phrases. It is not the grammar that shall secure the Doctor, nor weak analogies, where elegance comes in contest. To justify his expression he must have showed it usu tritum, or alleged the authority of some author of great note for it. I have not the leisure to examine his impertinent citations about those other compounds, nor yet of that simple verb duco; nay, to justify his saying he hath not brought one parallel example. He talks indeed very high, that duco, with its compounds, is a word of a large signification, and amongst the rest to bring, fetch, carry, &c. is so exceeding frequent in all authors, Plautus, Terence, Tully, Cæsar, Tacitus, Pliny, Seneca, Virgil, Horace, Ovid, Claudian, &c. that he must needs be either maliciously blind, or a very stranger to the Latin tongue, that doth not know it, or can have the face to deny it. I read, what will be my doom for not allowing his Latin; yet I must profess I dare secure the Doctor for having read all authors, notwithstanding his assertion, and I hope he will do the like for me. And for those which he hath read, had he brought no better proofs than these, he had, I am sure, been whipped soundly in Westminster School, for his impudence as well as ignorance, by the learned master thereof at present. But I dare further affirm, the Doctor hath not 408read in this point any, but only consulted with Robert Stephen’s Thesaurus Linguæ Latinæ, whence he hath borrowed his allegations in adduco; and for the other, I had not so much idle time as to compare them. And, lest the fact might be discovered, he hath sophisticated those authors whence Stephen cites the expressions, and imposed upon them others. If it be not so, or that the Doctor could not write it right when the copy was right before him, let him tell me where he did ever read in Plautus, adducta res in fastidium. I find the whole sentence in Pliny’s preface to Vespasian (out of whom in the precedent paragraph he cites it) about the middle: alia vero ita multis prodita, ut in factidium sint adducta, which is the very example Stephanus useth, although he doth premise his adducta res in fastidium. Let the Doctor tell where he ever did read in Horace, Ova noctuæ, &c. tædium vini adducunt. Did he, or any else, with the interposition of an &c. make Trochaics? I say, and Stephanus says so, too, that it is in Pliny, lib. xiii. cap. 15, near the end; the whole sentence runs thus: Ebriosis Ova noctuæ per triduum data in vino, tædium ejus adducunt. I doubt not but these are the places he aimed at, although he disguised and minced the quotations; if they be not, I should be glad to augment my Stephanus with his additions.

These things premised, I come to consider the Doctor’s proofs: Res eo adducta est: adducta vita in extremum: adducta res in fastidium: rem ad mucrones et manus adducere: contractares et adducta in augustum: res ad concordiam adduci potest: in ordinem adducerem: adducere febres, 409sitim, tedium vini (all in Robert Stephen) betwixt which and adducere malleum, what a vast difference there is, I leave them to umpire qui terretes et religiosas nacti sunt aures, who are the competent judges of elegancy, and only cast in the verdict of one or two, who are in any place (where the purity of the Latin tongue flourisheth) of great esteem. Losæus, in his Scopæ Linguæ Latinæ, ad purgandam Linguam a barbarie, &c. (would any think that the Doctor’s elegant expression, frequent in all authors, which none but the malicious or ignorant can deny, should suffer so contumelious an expurgation?) Losæus, I say, hath these words: Adferre plerique minus attenti utuntur pro adducere. Quod Plautus, in Pseudolo, insigni exemplo notat.

CA.--Attuli hunc.

PS.--Quid attulisti?

CA.--Adduxi volui dicere.

PS.--Quis istic est?

CA.--Charinus.

Satis igitur admonet discriminis inter ducere, reducere, adducere, et abducere, quæ de persona; et ferre, adferre, &c. quæ de re dicuntur. Idem, Demetrium, quem ego novi, adduce: argentum non moror quin feras. Cavendum igitur est ne vulgi more, (let the Doctor mark this, and know that this author is authentic amongst the Ciceronians), adferre de persona, dicamus, sed adducere; licet et hoc de certis quibusdam rebus non inepte dicatur. In this last clause he saith as much as Mr. Hobbes saith, and what the Doctor proves; but, that ever the Doctor brought an example which might resemble adducis malleum, is denied; for I have mentioned already his allegations, 410every one, of adduco. Another author, (a fit antagonist for the elegant Doctor), is the Farrago sordidorum Verborum, joined with the Epitome of L. Valla’s Elegancies. He saith: Accerse, adhuc Petrum, Latine dicitur, pro eo quod pueri dicunt, adfer Petrum. And this may suffice to justify Mr. Hobbes’s exception who proceeded no further than this author to tell the Doctor that adduco was used of animals. But the Doctor replies, this signification is true, but so may the other be also. I say if it never have been used so, it cannot be so, for we cannot coin new Latin words, no more than French or Spanish who are foreigners. Mr. Hobbes was upon the negative, and not to disprove the contrary opinion. If the Doctor would be believed, he must prove it by some example, (which is all the proof of elegancy), and till he do so, not to believe him, it is sufficient not to have cause. But, Doctor Wallis, why not adduco for a hammer as well as a tree? I answer yes, equally for either, and yet for neither. Did ever anybody go about to mock his readers thus solemnly? I do not find, to my best remembrance, any example of it in Stephen, and the Doctor is not wiser than his book; if there be, it is strange the Doctor should omit the only pertinent example, and trouble us with such impertinences for three or four pages. In Stephen there are adducere habenas and adducere lorum, but in a different sense. It is not impossible I may guess at the Doctor’s aim. In Tully de Nat. Deor. as I remember, there is this passage: Quum autem ille respondisset, in agro ambulanti ramulum adductum ut remissus esset, in oculum suum recidisse, where it signifies nothing 411else but to be bent, bowed, pulled back, and in that sense, the hammer of a clock, or that of a smith, when he fetcheth his stroke, may be said adduci. And this, I conceive, the Doctor would have us in the close think to have been his meaning; else, what doth he drive at in these words? “When you have done the best you can, you will not be able to find better words than adducere malleum and reducere, to signify the two contrary motions of the hammer, the one when you strike with it (excellently trivial!) the other when you take it back (better and better), What to do? to fetch another stroke. If any can believe that this was his meaning, I shall justify his Latin, but must leave it to him to prove its sense. If he intended no more, why did he go about to defend the other meaning, and never meddle with this? Which yet might have been proved by this one example of mine? May not, therefore, his own saying be justly retorted upon him in this case, Adducis malleum, ut occidas muscam?

Another exception is, Falsæ sunt, et multa istiusmodi (propositiones). I wish the Doctor could bring so good parallels, and so many, out of any author, for his Adducis malleum, as Tully affords in this case. Take one for all, out of the beginning of his Paradoxes: Animadverti sæpe Catonem, cum in senatu sententiam diceret, Locos graves ex Philosophia tractare, abhorrentes ab hoc usu forensi, et publico, sed dicendo consequi tamen, ut illa etiam populo probabilia viderentur. This is but a Solæcophanes, and hath many precedents more, as in the second book of his Academical Questions, &c.

412I cannot now stay upon each particular passage; I do not see any necessity of tracing the Doctor in all his vagaries. Now, he disallows tanquam diceremus, as if we should say. But why is that less tolerable than tanquam feceris, as if you had done? “It should be quasi, (forsooth!) or ac si, or tanquam si, which is Tully’s own word.” What is tanquam si become but one word? Tanquam si tua res agatur, &c. Good Doctor, leave out Tully and all Ciceronians, or you will for ever suffer for this, and your Adducis malleum. Is not this to put yourself on their verdict when you oppose Mr. Hobbes with Tully? But the Doctor gives his reason. And though he hath had the luck in his Adducis malleum, to follow the first part of that saying, Loquendum cum vulgo, yet now it is, sentiendum cum sapientibus. For tanquam without si signifies but as, not as if. It is pity the Doctor could not argue in symbols too, that so we might not understand him; but suppose all his papers to carry evidence with them, because they are mathematically scratched. How does he construe this:--

“Plance tumes alto Drusorum sanguine, tanquam

Feceris ipse aliquid, propter quod nobilis esses.”

So Cœlius, one much esteemed by Cicero, who hath inserted his Epistles into his works, saith, in his fifth Epistle (Tul. Epist. Fam. lib. viii. ep. 5), Omnia desiderantur ab eo tanquam nihil denegatum sit ei quo minus paratissimus esset qui publico negotio præpositus est. But it was not possible the Doctor should know this, it not being in Stephen, where his examples for tanquam si are.

But, the Doctor having pitched upon this criticism, and penned it, somebody, I believe, put him 413in mind of the absurdity thereof; and yet the generous Professor, (who writes running hand and never transcribed his papers, if I am not misinformed), presumed nobody else could be more intelligent than he, who had perused Stephen. He would not retract anything, but subjoins, “That he will allow it as passable, because other modern writers, and some of the ancients, have so used it, as Mr. Hobbes hath done.” I know not what authors the Doctor meant, for, if I am not much mistaken, I do not find any in Stephen. His citation of Columella is not right, (lib. v. cap. 5), nor can I deduce anything thence till I have read the passage, but, if he take Juvenal and Cœlius for modern authors, I hope he will admit of Accius, Nævius, and Carmenta, for the only ancients. Let him think upon this criticism, and never hope pardon for his Adducis malleum, which is not half so well justified, and yet none but madmen or fools reject it.

But certainly the Doctor should not have made it his business to object Anglicisms, in whose Elenchus I doubt not but there may be found such phrases as may serve to convince him that he is an Englishman, however Scottified in his principles. If the Doctor doubt of it, or but desire a catalogue, let him but signify his mind, and he shall be furnished with a Florilegium. But I am now come to the main controversy about Empusa. The Doctor saith nothing in defence of his quibble, nor gives any reason why he jumbled languages to make a silly clinch, which will not pass for wit either at Oxford or at Cambridge; no, nor at Westminster.

It seems he had derived Empusa from ἓν and ποῦς, 414and said it was a kind of Hobgoblin that hopped upon one leg: and hence it was that the boys’ play (Fox come out of thy hole) came to be called Empusa. I suppose he means Ludus Empusæ. This derivation he would have to be good, and that we may know his reading, (though he hath scarce consulted any of the authors), he saith Mr. Hobbes did laugh at it, until somebody told him that it was in the Scholiast of Aristophanes (as good a critic as Mr. Hobbes), Eustathius, Erasmus, Cœlius Rhodiginus, Stephanus, Scapula, and Calepine. But sure he doth not think to scape so. To begin with the last; Calepine doth indeed say, uno incedit pede, unde et nomen. But he is a Modern, and I do not see why his authority should outweigh mine if his author’s reasons do not. He refers to Erasmus and Rhodiginus. Erasmus in the adage, Proteo mutabilior hath these words of Empusa: Narrant autem uno videri pedi--this is not to hop--unde et nomen inditum putant, Ἔμπουσαν ὁιονεὶ ἑνίποδα. He doth not testify his approbation of the derivation at all, only lets you know what etymologies some have given before him. And doth anybody think that Dr. Harmar was the first which began to show his wit, (or folly), in etymologizing words? Cœlius Rhodiginus doth not own the derivation, only saith, Nominis ratio est, ut placet Eustathio, quia uno incedit pede;--is this to hop?--sed nec desunt qui alterum interpretentur habere æneum pedem, et inde appellatam Empusam; quod in Batrachis Aristophanes expressit. And then he recites the interpretation that Aristophanes’s Scholiast doth give upon the text, of which by and by. If any credit be to be attributed to this allegation, his last 415thoughts are opposite to Dr. Wallis; and Empusa must be so called, not because she hopped upon one leg, but because she had but one, the other being brass. But for the former derivation he refers to Eustathius.

As to Eustathius, I do easily conjecture that the reader doth believe that Rhodiginus doth mean Eustathius upon Homer, for that is the book of most repute and fame, his other piece being no way considerable for bulk or repute. But it is not that book, nor yet his History of Ismenias, but his notes upon the 725th verse of Dionysius Περῖηγητής. The poet had said of the stone Jaspis, that it was

Ἐχθρηὶν Ἐμπούσησι καὶ ἄλλοις ἔιδώλοισιν,

Upon which Eustathius thus remarks: Δοκεῖ γαρ ἀλεξίκακος εἶναι ἡ λίθος ἅυτη, καὶ ἀποτροπιαςτικὴ φασμἀτων, ὧν ἕν ἐςτι καὶκαὶ ἡ Ἔμπουσα, δαιμονιόν τι τερί τἰὼ Ἑκάτην, ἑνὶ ποδὶ δοκοῦν δἰήκεσθαι· (fortè διερείδεσθαι Steph.) ὄθεν καὶ παρονομάζεται, ὡς ἔι τις ἔιπη μονόπους ποδι ζωοῦ· ὡς τοῦ ἑτέρου ποδος χαλκοῦ ὄντος, κατὰ τὸν μῦθον. This testimony doth not prove anything of hopping, and, as to the derivation, I cannot but say that Eustathius had too much of the grammarian in him, and this is not the first time, neither in this book, nor elsewhere, wherein he hath trifled. It is observable out of the place, that there were more Empusas than one, as, indeed, the name is applied by several men to any kind of frightful phantasm. And so it is used by several authors, and for as much as phantasms are various, according as the persons affrighted have been severally educated, &c. every man did impose this name upon his own apprehensions. This gave men occasion to fain Empusa as such--for who will believe that she was not apprehended as having four legs, when she 416appeared in the form of a cow, dog, &c.--but, as apprehended by Bacchus and his man at that time. I do not find that she appeared in any shape but such as made use of legs in going, whence I imagine that Empusæ might be opposite to the θεοὶ νεποδες, which appellation was anciently fixed upon the gods, (propitious) upon a two-fold account; first, for that they were usually effigiated as having no feet, which is evident from ancient sculpture, and secondly, for that they are all said not to walk, but rather swim, if I may so express that non gradiuntur, sed fluunt, which is the assertion of all the commentators I have ever seen upon that verse of Virgil:--

“Et vera incessu patuit dea”----

This whole discourse may be much illustrated from a passage in Heliodorus, Æthiop. lib. iii. sec. 12, 13. Calasiris told Cnemon that the Gods Apollo and Diana did appear unto him; Cnemon replied, Ἀλλὰ τίνα δὴ τρόπον ἒφασκες ἐνδεδεῖχθαἱ σοι τοῦς θεοῦς ὅτι μὴ ἐνύπνιον ἦλθον, ἀλλ’ ἐναργῶς ἐφᾶνησαν; upon this the old priest answered, that both gods and demons, when they appear to men, may be discovered by the curious observer, both in that they never shut their eyes, καὶ τῳ βαδίσματι πλέον, οὐ κατὰ διάστησιν τῶν ποδῶν οὐδέ μετάθεσιν ἀνυομένω, ἀλλὰ κατὰ τινα ρὕμην ἀέριον, καὶ ὁρμὴν ἀπαραπόδιστον, τεμνόντων μᾶλλον τὸ περιεχον ἢ διαπορευομένων. Δὶο δὴ καὶ τὰ ἀγάλματα τῶν θεῶν ἈιγύπτιοιἈιγύπτιοι τὼ πὸδέ ζευγνύντες καὶ ὥσπερ ἑνοῦντες ἵστᾶσιν. ἅ δὴ καὶ Ὅμηρος ἐιδῶς, ἅτε Ἀιγύπτιος, καὶκαὶ τὴν ἱερὰν πάιδευσιν ἐκδιδαχθείς, συμβολικῶς τοῖς ἔπεσιν ἐναπεθετο, τοῖς δυναμένοις συνιέναι γνωριζειν καταλιπών, ἐπι τοῦ ποσειδῶνος, το

Ἴχνια γὰρ μετόπισθε, ποδῶν ἠδέἠδέ κνημάων

Ῥεῖ ἔγνων ἀπιὸντος.

οἴον ῥέοντος ἐν τῆ πορεία, τοῦτο γάρ εστι τό ῥεῖ ἀπιόντος, καὶ οῦχ ὥς 417τινες ἠπάτηνται, ῥᾳδίως ἔγνωνῥᾳδίως ἔγνων ὑπολαμβάνοντες. Farnaby, upon the place in Virgil, observes, that Deorum incessus est continuus et æqualis, non dimotis pedibus, neque transpositis, ἀλλὰ κατὰ ῥύμην ἀέριονἀέριον. Cornelius Schrevelius in the new Leyden notes saith, Antiquissima quæque Deorum simulachra, quod observarunt viri magni, erant τοῦς πόδας συμβεβηκότα, diique ipsi non gradiuntur sed fluunt. Their statues were said to stand rather upon columns than upon legs, for they seem to have been nothing but columns shaped out into this or that figure, the base whereof carrying little of the representation of a foot. These things being premised, I suppose it easy for the intelligent reader to find out the true etymology of Empusa, quasi ἐν ποσιν οῦσα, or βάινουσα, from going on her feet, whereas the other gods and demons had a different gait. If any can dislike this deduction, and think her so named from ἑνιπους, whereas she always went upon two legs, (if her shape permitted it) though she might draw the one after her, as a man doth a wooden leg: I say, if any, notwithstanding what hath been said, can join issue with the Doctor, my reply shall be Σοὶ μὲν ταῦτα δοκοῦντ’ ἐστὶν, ἐμὸι δὲ τάδε.