Chapter X (9)

I come now to Stephen, who, in his index, and in the word ποδίζω, gives the derivation of _Empusa_. Ποδιζω, _gradior, incedo_, (not to hop) _sic Suidas_ Ἔμπουσαν _dictam ait_ παρὰ το ἑνὶ ποδιζειν. In the index thus: _sunt qui dictam putent_ παρὰ τὸ ἑνὶ ποδὶζειν, _quod uno incedat pedi, quasi_ Ἔμπουσαν, _alterum enim pedem æneum habet_. But neither Stephen, nor any else, except _Suidas_, whom the hypercritical Doctor had not seen, no, not the Scholiast of Aristophanes (a better critic than Mr. Hobbes) doth relate the etymology as their own. Nay, there is not one that saith _Empusa_ hopped on one leg, which is to be proved out of them. The great Etymological Dictionary deriveth it παρὰ τὸ ἐμποδιζειν, to _hinder_, _let_, &c. its apparition being a token of ill luck. But, as to the Doctor’s deduction, it saith, Ἔμπουσα Ψιλοῦπαι, εἰ καὶ δοκεῖ παρὰ τὸ ἕνα συγκεῖσθαι. It doth only _seem_ so. And it is strange that ἑν should not alter only its _aspiration_, but change its ν into μ, which I can hardly believe admittable in Greek, least there should be no difference betwixt its derivatives and those of ἐν. When I consider the several μορμόνες which the Grecians had, some whereof did fly, some had no legs, &c., I can think that the origin of this name may have been thus: some amazed person saw a _spectrum_, and, giving another notice of it, his companion might answer, it is Βριμὼ, Μορμὼ Ἡκὰτη, but he, meeting with a new phantasm, cries, ἐν ποσὶ βαίνει or βαδίζει, for which apprehension of his, somebody coined this expression of Ἔμποῦσα. It may also be possibly deduced from Ἐμποδὶζω, so that τύχη ἐμποδιζουσα might afterwards be reduced to the single term of _Empusa_. Nor do I much doubt but that those who are conversant in languages, and know how that several expressions are often jumbled together to make up one word upon such like cases, will think this a probable origination. I believe, then, that Mr. Hobbes’s friend did never tell him it was in Eustathius, or that _Empusa_ was an _hopping phantasm_. It had two legs and went upon both, as a man may upon a wooden leg. Ἔμποῦσα is also a name for Lamia, and such was that which Menippus might have married, which, I suppose, did neither hop nor go upon one leg, for he might have discovered it. But Mr. Hobbes did not except against the derivation, (although he might justly, derivations made afterwards carrying more of fancy than of truth, and the Doctor is not excused for asserting what others barely relate, none approve), but asked him where that is, in what authors _he read that boys’ play to be so called_. To which question, the Doctor, to show his reading and the good authors he is conversant in, replies, _in Junius’s Nomenclator, Rider and Thomas’s Dictionary, sufficient authors in such a business_, which, methinks, no man should say that were near to so copious a library. It is to be remembered that the trial now is in Westminster School, and amongst Ciceronians, neither whereof will allow those to be sufficient authors of any Latin word. Alas, they are but _Vocabularies_; and, if they bring no author for their allegation, all that may be allowed them is, that, by way of allusion, our modern play may be called _Ludus Empusæ_. But that it is so called we must expect, till some author do give it the name. These are so good authors, that I have not either of them in my library. But I have taken the pains to consult, first, Rider; I looked in him, (who was only author of the English Dictionary) and I could not find any such thing. It is true, in the Latin Dictionary, which is joined with Rider, but made by Holyoke; (O that the Doctor would but mark!) in the index of obsolete words, there is _Ascoliasmus, Ludus Empusæ_, _Fox to thy hole_, for which word, not signification, he quoteth Junius. The same is in Thomasius, who refers to Junius in like manner. But could the Doctor think the word obsolete, when the play is still in fashion? Or, doth he think that this play is so ancient as to have had a name so long ago, that it should now be grown obsolete? As for Junius’s interpretation of _Empusa_, it is this: _Empusa, spectrum, quod se infelicibus ingerit, uno pede ingrediens_. Had the Doctor ever read him, he would have quoted him for his derivation of _Empusa_, I suppose. In Ascoliasmus, he saith, _Ascoliasmus, Empusæ Ludus, fit ubi, altero pede in aere librato, unico subsiliunt pede:_ ἀσκολιασμὸς _Pollux; Almanicè, Hinckelen; Belgicè,_ _Op een been springhen; Hinckepincken, Flandris_. But what is it in English he doth not tell, although he doth so in other places often. What the Doctor can pick out of the Dutch I know not; but, if that do not justify him, as I think it doth not, he hath wronged Junius, and greatly imposed upon his readers.

But, to illustrate this controversy further, I cannot be persuaded the Doctor ever looked into Junius, for, if he had, I am confident, according to his wonted accurateness, he would have cited Pollux’s _Onomasticon_ into the bargain, for Junius refers to him, and I shall set down his words, that so the reader may see what _Ascoliasmus_ was, and all the Doctor’s authors say _Ludus Empusæ_ and _Ascoliasmus_ were one and the same thing. Julius Pollux (lib. ix. cap. 7): Ὁ δε Ἀσκολὶασμὸς, (old editions read it, Ἀ’σκολιασμὸς et ασκολιάζω) τοῦ ἑτέρου ποδὸς αἰωρουμένου, κατὰ μόνου τοῦ ἑτέρου πηδᾶν ἔπόιει; ὅπερ Ἀσκωλιάζὲιν ὠνόμαζον· ἤτοι εἰς μἢκος ἐνήλλαντο, ἢ ὁ μὲν ἐδίωκεν οὕτως, οἱ δὲ ὑπέφευγον ἐπ’ ἀμφοῖν θὲοντες, ἕως τινὸς τῳ φερομένῳ ποδὶ ὁ διὼκων δυνηθῇ τυχεῖν· ἤ καὶ στάντες ἐπήδων, ἀριθμοῦντες τὰ πηδήματα· προσέκειτο γὰρ τῷ πλήθει τὸ νικᾶν. Ἀσκωλιάζειν δὲ ἐκαλεῖτο καὶ τὸ ἐπιπηδᾶν ἀσκῷ κενῷ καὶ ὑποπλέω πνευματος, ἠλείμμένω, ἵναπερ ὀλισθάνοιεν περὶ τὴν ἀλοιφὴν. “So that _Ascoliasmus_, and consequently, _Ludus Empusæ_, was a certain sport which consisted in hopping, whether it were by striving who could hop furthest, or whether only one did pursue the rest hopping, and they fled before him on both legs, which game he was to continue till he had caught one of his fellows, or whether it did consist in the boys’ striving who could hop longest. Or, lastly, whether it did consist in hopping upon a certain bladder, which, being blown up and well oiled over, was placed upon the ground for them to hop upon, that so the unctuous bladder might slip from under them and give them a fall.” And this is all that Pollux holds forth. Now, of all these ways, there is none that hath any resemblance with our _Fox to thy hole_; but the second: and yet, in its description, there is no mention of beating him with gloves, as they do now-a-days, and wherein the play consists as well as in hopping. It might, notwithstanding, be called _Ludus Empusæ_, but not in any sort our _Fox to thy hole_; so that the Doctor and his authors are out, imposing that upon Junius and Pollux which they never said. And thus much may suffice as to this point. I shall only add out of Meursius’s _Ludi Græci_, that _Ascolia_ were not _Ludus Empusæ_ but _Bacchisacra_, and he quotes Aristophanes’s Scholiast in Plutus, Ἀσκώλια ἑορτὴ Διονύσου ἀσκὸν γαρ οἵνου πληροῦντες, ἑνὶ ποδὶ τοῦτον ἐπεπήδον, καὶ ὁ πηδήσας ἆθλον εἶχε τὸν οἵνου. As also Hesychius, Ἀσκωλιάζειν, κυρίως τὸ ἐπὶ τοῦς ἀσκοὺς ἅλλεσθαι.

But I could have told the Doctor where he might have read of _Empusa_ as being the name of a certain sport or game, and that is, _in Turnebus Adversaria_, lib. xxvii. cap. 33. There he speaks of several games mentioned by Justinian in his _Code_, at the latter end of the third book, one of which he takes to be named _Empusa_; adding withal, _that the other are games, it is indisputable_, only _Empusa in lite et causa erit, quod nemo nobis facile assensurus sit Ludum esse, cum constet spectrum quoddam fuisse formas, varie mutans. Sed quid vetat eo nomine Ludum fuisse? Certe ad vestigia vitiatæ Scripturæ quam proximo accedit._ Yet he only is satisfied in this conjecture, till somebody else shall produce a better. And now what shall I say? Was not Turnebus as good a critic, and of as great reading as Dr. Wallis, who had read over Pollux, and yet is afraid that nobody will believe _Empusa_ to have been a game, and all he allegeth for it is, _quid vetat_? Truly, all I shall say, and so conclude this business, is, that he had read over an infinity of books, yet, had not had the happiness, which the Doctor had, to consult with _Junius’s Nomenclator, Thomasius and Rider’s Dictionary, authors sufficient in such a case_.

I now come to the Doctor’s last and greatest triumph, at which I cannot but stand in admiration, when I consider he hath not got the victory. Had the Doctor been pleased to have conversed with some of the fifth form in Westminster School, (for he needed not to have troubled the learned master), he might have been better informed than to have exposed himself thus.

Mr. Hobbes had said that στιγμὴ signified _a mark with a hot iron_; upon which saying the Doctor is pleased to play the droll thus: “Prithee tell me, good Thomas, before we leave this point, (O the wit of a divinity doctor!) who it was told thee that στιγμὴ was a mark with an hot iron, for it is a notion I never heard till now, and do not believe it yet. Never believe him again that told thee that lie, for as sure as can be, he did it to abuse thee; ϛιγμὴ signifies a distinctive point in writing, made with a pen or quill, not a mark made with a _hot iron_, such as they brand rogues withal; and, accordingly, ϛιζω δῖαϛιζω, _distinguo_, _interstinguo_, are often so used. It is also used of a _mathematical_ point, or somewhat else that is very small, στιγμὴ χρὸνου, a moment, or the like. What should come in your cap, to make you think that ϛῖγμὴ signifies a mark or brand with a _hot iron_? I perceive where the business lies; it was ϛίγμα ran in your mind when you talked of ϛιγμὴ, and, because the words are somewhat alike, you jumbled them both together, according to your usual care and accurateness, as if they had been the same.”

When I read this I cannot but be astonished at the Doctor’s confidence, and applaud him who said, ἀμάθεια θάρσὸς φέρει. That the Doctor should never hear that ϛιγμὴ signifies _a mark with a hot iron_, is a manifest argument of his ignorance. But, that he should advise Mr. Hobbes not to believe his own readings, or any man’s else that should tell him it did signify any such thing, is a piece of notorious impudence. That ϛιγμὴ _signifies a distinctive point in writing made with a pen or quill_, (is a pen one thing and a quill another to write with?) nobody denies. But, it must be withal acknowledged it signifies many things else. I know the Doctor is a _good historian_, else he should not presume to object the want of history to another; let him tell us how long ago it is since men have made use of pens or quills in writing; for, if that invention be of no long standing, this signification must also be such, and so it could not be that from any allusion thereunto the mathematicians used it for a point. Another thing I would fain know of this great historian, how long ago ϛίζω and διαϛίζω began to signify _interpungo_? For, if the mathematics were studied before the mystery of printing was found out, (as shall be proved whenever it shall please the Doctor, out of his no reading, to maintain the contrary), then the _mathematical_ use thereof should have been named before the _grammatical_. And, if this word be translatitious, and that sciences were the effect of long contemplation, the names used wherein are borrowed from talk, Mr. Hobbes did well to say, that στιγμὴ precedaneously to that _indivisible_ signification which it afterwards had, did signify a _visible mark_ made by a hot iron, or the like. And, in this procedure, he did no more than any man would have done, who considers that all our knowledge proceeds from our senses; as also that words do, _primarily_, signify things obvious to _sense_, and only _secondarily_, such as men call _incorporeal_. This leads me to a further consideration of this word. Hesychius, (of whom it is said that he is _Legendus non tanquam Lexicographus, sed tanquam justus author_), interprets στιγμὴ, νυγμή, which is a point of a greater or lesser size, made with any thing. So ϛίζω signifies to prick or mark with anything in any manner, and hath no impropriated signification in itself, but according to the writer that useth it. Thus, in a _grammarian_ ϛίζω signifies to _distinguish_, by _pointing_ often; sometimes, even in them, it is the same with ὀβελίζω; sometimes it signifies to set a mark that something is wanting in that place, which marks were called ϛιγμαί. In matters of policy, ϛίζω signifies to _disallow_, because they used to put a ϛιγμὴ (not ϛίγμα) before his name who was either disapproved or to be mulcted. In punishment it signifies to _mark_ or _brand_, whereof I cannot at present remember any other ways than that of an _hot iron_, which is most usual in authors, because most practised by the ancients. But, that the mark which the _Turks_ and others do imprint without burning may be said ϛίζεσθαι, I do not doubt, no more than that Herodian did to give that term to the ancient Britons, of whom he says, τὰ σώματα ἐϛίζοντο γραφαῖς ποικίλαις, καὶ ζώων παντοδαπῶν εἰκόσι. Thus, horses that were branded with κάππα and σαν (κοππἀτιαι and σαμφοραι) were said ϛίζεσθαι. Thus, in its origin, ϛιγμὴ doth signify a _brand or mark with an hot iron_, or the like; and that must be the proper signification of στιγμὴ, which is proper to ϛίζω, none but such as Dr. Wallis can doubt. In its _descendants_ it is no less evident, for, from στιγμὴ comes _stigmosus_, which signifies to be branded; _Vitelliana cicatrice stigmosus_, not _stigmatosus_. So Pliny in his Epistles, as Robert Stephen cites it. And στιγματιας (the derivative of στιγμὴ, which signifies any mark, as well as a brand, even such as remain after stripes, being black and blue), was a nickname imposed upon the grammarian Nicanor, ὅτι περὶ στιγμῶν ἐπολυλόγησε. And, though we had not any examples of στῖγμὴ being used in this sense, yet, from thence, for any man to argue against it, (but he who knows no more than Stephen tells him) is madness, unless he will deny that any word hath lost its right signification, and is used only, by the authors we have, although neither the Doctor nor I have read all them, in its analogical signification. I have always been of opinion, that στιγμὴ signified a _single point_, big or little, it matters not; and στίγμα, a _composure of many_; as γραμμὴ signifies a _line_, and γράμμα a _letter_, made of several lines. For στίγμα signified the _owl_, the _sæmæna_, the letter K, yea, _whole words, lines, epigrams_ engraven in men’s faces; and στιγμὴ, I doubt not, had signified _a single point_, had such been used, and so it became translatitiously used by grammarians and mathematicians. I could give grounds for this conjecture, and not be so impertinent as the Doctor in his sermon, where he told men that σοφός was not in Homer; that from ἄφρων came _ebrius_; that _sobrietas_ was not bad Latin, and that _sobrius_ was once, as I remember, in Tully. Is this to speak suitably to the oracles of God, or rather to lash out into idle words? Hath the Doctor any ground to think these are not impertinences? Or, are we, poor mortals, accountable for such _idle_ words as fall from us in private discourses, whilst these ambassadors from heaven _droll_ in the pulpit without any danger of an after-reckoning?

But I proceed to a further survey of the Doctor’s intolerable ignorance. His charge in the end of the _school-master’s_ rant is, that he should _remember_ στίγμα and στιγμὴ _are not all one_. I complained before that he hath not cited Robert Stephen aright; now I must tell him he hath been negligent in the reading of Henry Stephen: for in him he might have found that στίγμα was sometimes all one with στιγμὴ, though there be no example in him wherein στιγμὴ is used for στίγμα. Hath not Hesiod, (as Stephen rightly citeth it), in his _Scutum_, 166-67.

Στἴγματα δ’ ὥς ἐπέφαντο ἴδεῖν δεινοῖσι δράκουσι
Κυανέα κατὰ νῶτα

_ubi scholiastes_ ὥσπερ δὲ στιγμαὶ ἦσαν ἐπάνω, τῶν ῥάχεων τῶν δρακόντων, κατάστίκτοι γὰρ καὶ ποικίλοι ὁι ὄφεις. So Johannes Diaconus upon the place, a man who (if I may use the Doctor’s phrase) was _as good a critic as_ the Geometry Professor.

Thus much for the _Doctor_. To the understanding _reader_, I say that στιγμὴ is used for burning with a hot iron: _2 Macchab._ ix. 11, where speaking of Antiochus’s lamentable death, his body putrefying and breeding worms, he is said, ἐις ετίγνωσιν τοῦ θεοῦ ἔρχεθαι θείᾳ μάστιγι, κατα στιγμὴν ἐπιτεινόμενος ταῖς ἀλγηδόσι; _being pained as if he had been pricked or burned with hot irons_. And that this is the meaning of that elegant writer, shall be made good against the Doctor, when he shall please to defend the vulgar interpretation. Pausanias, in _Bœoticis_, speaking of Epaminondas, who had taken a town belonging to the Sicyonians, called Phœbia (Φουβία) wherein were many Bœotian fugitives, who ought, by law, to have been put to death, saith he dismissed them under other names, giving them only a _brand_ or _mark_. Πόλισμα ἑλὼν Σικυωνἰων Φουβίαν, ἔνθὰ ἦσαν το πολὺ οἱ Βοιώτιοι φυγάδες, στιγμήν ἀφίησι τοῦς ἐγκαταληφθέντας ἄλλην σφίσιν ἣν ετυχε πατρίδα ἐπονομάζων ἐκάστω. It is true στιγμὴν is here put _adverbially_, but that doth not alter the case. Again, Zonaras, in the third tome of his History, in the life of the Emperor Theophilus, saith, that when Theophanes and another monk had reproved the said emperor for demolishing images, he took and _stigmatized_ each of them with twelve _iambics_ in their faces: εἶτα καὶ τὰς ὄψεις ἀυτῶν κάτεστιξε καὶ ταῖς στιγμαῖς μέλαν ἐπέχεε γράμματα δὲ ἐτύπουν τὰ στιγματα, τὰ δὲ ἦσαν ἴαμβοι οὗτοι. A place so evident, that I know not what the Doctor can reply. This place is just parallel to what the same author saith in the life of Irene, τἀς ὄψέις σφών καταστιξας ἐν γράμμασι, μέλανος εγχεομένου τοῖς στίγμασι. If the Doctor object that he is a modern author, he will never be able to render him as inconsiderable as Adrianus Junius’s _Nomenclator_, Thomasius and Rider. If any will deny that he writes good Greek, Hieronymus Wolfius will tell them, his only fault is περισσολογια, _redundancy_ in words, and not the use of _bad_ ones.

Another example of στιγμὴ used in this sense, is in the collections out of Diodorus Siculus, lib. xxxiv. as they are to be found at the end of his works, and as Photius hath transcribed them into his _Bibliotheca_. He saith that the Romans did buy multitudes of servants and employ them in Sicily: Οἷς, ἐκ τῶν σωματοτροφείων ἀγεληδὸν απαχθεῖσιν, ἐυθύς χαρακτῆρα ἐπέβαλλον, καὶ στιγμὰς τοἴς σώμασιν. These are the words but of one author, but ought to pass for the judgment of two, seeing Photius, by inserting them, hath made them his own.

Besides, it is the judgment of a great _master_ of the Greek tongue, that _stigmata non tam puncta ipsa quam punctis variatam superficiem Græci vocaverunt_. I need not, I suppose, name him, so great a critic as the Doctor cannot be ignorant of him.

Nor, were στίγματα commonly, but upon extraordinary occasions, imprinted with an hot iron. The letters were first made by incision, then the blood _pressed_, and the place filled up with ink, the composition whereof is to be seen in Aetius. And thus they did use to _matriculate_ soldiers also in the hand. Thus, did the Grecian emperor, in the precedent example of Zonaras. And if the Doctor would more, let him repair to Vinetus’s comment upon the fifteenth Epigram of Ausonius.

And now I conceive enough hath been said to vindicate Mr. Hobbes, and to show the insufferable ignorance of the puny professor, and unlearned critic. If any more shall be thought necessary, I shall take the pains to collect more examples and authorities, though I confess I had rather spend time otherwise, than in matter of so little moment. As for some other passages in his book, I am no competent judge of _symbolic stenography_. The Doctor (Sir Reverence) might have used a cleanlier expression than that of a _shitten piece_, when he censures Mr. Hobbes’s book.

Hitherto the letter.[1] By which you may see _what came into my (not square) cap to call_ στιγμὴ _a mark with a hot iron, and that they who told me_ that, did no more tell me a lie than they told you a lie that said the same of στίγμα; and, if στιγμὴ be not right as I use it now, then call these notes not στιγρας, but στίγματα. I will not contend with you for a trifle. For, howsoever you call them, you are like to be known by them. Sir, the calling of a divine hath justly taken from you some time that might have been employed in geometry. The study of algebra hath taken from you another part, for algebra and geometry are not all one; and you have cast away much time in practising and trusting to symbolical writings; and for the authors of geometry you have read, you have not examined their demonstrations to the bottom. Therefore, you perhaps may be, but are not yet, a geometrician, much less a good divine. I would you had but so much ethics as to be civil. But you are a notable critic; so fare you well, and consider what honour you do, either to the University where you are received for professor, or to the University from whence you came thither, by your geometry; and what honour you do to Emanuel College by your divinity; and what honour you do to the degree of Doctor, with the manner of your language. And take the counsel which you publish out of your encomiast his letter; think me no more worthy of your pains, you see how I have fouled your fingers.

-----

Footnote 1:

Written by Henry Stubbe, M.A. of Christ Church, Oxford, who was,
according to Anthony a Wood, “the most noted personage of his age that
these late times have produced.”

-----

THREE PAPERS

PRESENTED TO THE ROYAL SOCIETY

AGAINST DR. WALLIS.

TOGETHER WITH

CONSIDERATIONS

ON DR. WALLIS’S ANSWER TO THEM,

BY

THOMAS HOBBES,

OF MALMESBURY.

THREE PAPERS

PRESENTED TO THE ROYAL SOCIETY.

==========

TO THE RIGHT HONOURABLE AND OTHERS, THE LEARNED MEMBERS OF THE ROYAL
SOCIETY, FOR THE ADVANCEMENT OF SCIENCES.

PRESENTETH _to your consideration, your most humble servant, Thomas Hobbes, (who hath spent much time upon the same subject), two propositions, whereof the one is lately published by Dr. Wallis, a member of your Society, and Professor of Geometry; which if it should be false, and pass for truth, would be a great obstruction in the way to the design you have undertaken. The other is a problem, which, if well demonstrated, will be a considerable advancement of geometry; and though it should prove false, will in no wise be an impediment to the growth of any other part of philosophy._

DR. WALLIS,
DE MOTU, _Cap._ v. _Prop._ 1.

If there be understood an infinite row of quantities beginning with 0 or (1)/(0), and increasing continually according to the natural order of numbers, 0, 1, 2, 3, &c. or according to the order of their squares, as, 0, 1, 4, 9, &c. or according to the order of their cubes, as, 0, 1, 8, 27, &c. whereof the last is given; the proportion of the whole, shall be to a row of as many, that are equal to the last, in the first case, as 1 to 2; in the second case, as 1 to 3; in the third case, as 1 to 4, &c.

This proposition is the ground of all his doctrine concerning the centres of gravity of all figures. Wherein may it please you to consider:

First, whether there can be understood an infinite row of quantities, whereof the last can be given. Secondly, whether a finite quantity can be divided into an infinite number of lesser quantities, or a finite quantity can consist of an infinite number of parts, which he buildeth on as received from Cavallieri. Thirdly, whether (which in consequence he maintaineth) there be any quantity greater than infinite. Fourthly, whether there be, as he saith, any finite magnitude of which there is no centre of gravity. Fifthly, whether there be any number infinite. For it is one thing to say, that a quantity may be divided perpetually without end, and another thing to say, that a quantity may be divided into an infinite number of parts. Sixthly, if all this be false, whether that whole book of _Arithmetica Infinitorum_, and that definition which he buildeth on, and supposeth to be the doctrine of Cavallieri, be of any use for the confirming or confuting of any propounded doctrine.

Humbly praying you would be pleased to declare herein your judgment, the examination thereof being so easy, that there needs no skill either in geometry, or in the Latin tongue, or in the art of logic, but only of the common understanding of mankind to guide your judgment by.

THOMAS HOBBES,
ROSET. _Prop._ v.

_To find a straight line equal to two-fifths of the arc of a
quadrant._

I describe a square A B C D, and in it a quadrant D A C. Suppose D T be two-fifths of D C, then will the quadrantal arc T V be two-fifths of the arc C A. Again let D R be a mean proportional between D C and D T; then will the quadrantal arc R S be a mean proportional between the arc C A and the arc T V.

Suppose further a right line were given equal to the arc C A, and a quadrantal arc therewith described; then will D C, C A, the arc on C A be continually proportional. Set these proportionals in order by themselves.

D C, C A, arc on C A∺
D R, R S, arc on R S∺
D T, T V, arc on T V∺

which are in continual proportion of the semi-diameter of the arc. And D C, D R, D T are in a continual proportion by construction, and therefore also C A, R S, T V, and arc on C A, arc on R S, arc on T V, in continual proportion.

Therefore as D C to R S, so is R S to the arc on T V. And D C, R S, the arc on T V will be continually proportional. And because D C, C A, the arc on C A are also continually proportional, and have the first antecedent D C common; the proportion of the arc on C A to the arc on T V is (by Eucl. xiv. 28) duplicate of the proportion of C A to R S, and the arc on R S a mean proportional between the arc on C A and the arc on T V.

Now if D C be greater than R S, also R S must be greater than the arc on T V; and the arc C A greater than the arc on R S. Therefore seeing D C, C A, arc on C A, are continually proportional; the arc on T V, the arc on R S, the arc on C A cannot be continually proportional, which is contrary to what has been demonstrated. Therefore D C is not greater than R S. Suppose, then, R S to be greater than D C, then will the arc on R S be a mean proportional between the arc on T V, and a greater arc than that on C A; and so the inconvenience returneth. Therefore the semidiameter D C is equal to the arc R S, and D R equal to T V, that is to say to two-fifths of the arc C A, which was to be demonstrated. Nor needeth there much geometry for examining of this demonstration. Therefore I submit them both to your censure, as also the whole _Rosetum_, a copy whereof I have caused to be delivered to the secretary of your society.

TO THE

RIGHT HONOURABLE AND OTHERS,

THE LEARNED MEMBERS

OF

THE ROYAL SOCIETY,

FOR THE ADVANCEMENT OF THE SCIENCES.

-------

Presenteth to your consideration, your most humble servant Thomas Hobbes, a confutation of a theorem which hath a long time passed for truth; to the great hinderance of Geometry, and also of Natural Philosophy, which thereon dependeth.

THE THEOREM.

_The four sides of a square being divided into any number of equal parts, for example into 10; and straight lines drawn through the opposite points, which will divide the square into 100 lesser squares; the received opinion, and which Dr. Wallis commonly useth, is, that the root of those 100, namely 10, is the side of the whole square._

THE CONFUTATION.

_The root 10 is a number of those squares, whereof the whole containeth 100, whereof one square is an unity; therefore the root 10, is 10 squares: Therefore the root of 100 squares is 10 squares, and not the side of any square; because the side of a square is not a superficies, but a line. For as the root of 100 unities is 10 unities, or of 100 soldiers 10 soldiers: so the root of 100 squares is 10 of those squares. Therefore the theorem is false; and more false, when the root is augmented by multiplying it by other greater numbers._

Hence it followeth, that no proposition can either be demonstrated or confuted from this false theorem. Upon which, and upon the numeration of infinites, is grounded all the geometry which Dr. Wallis hath hitherto published.

And your said servant humbly prayeth to have your judgment hereupon: and that if you find it to be false, you will be pleased to correct the same: and not to suffer so necessary a science as geometry to be stifled, to save the credit of a professor.

TO THE

RIGHT HONOURABLE AND OTHERS,

THE LEARNED MEMBERS

OF

THE ROYAL SOCIETY,

FOR THE ADVANCEMENT OF THE SCIENCES.

---

Your most humble servant Thomas Hobbes presenteth, that the quantity of a line calculated by extraction of roots is not to be truly found. And further presenteth to you the invention of a straight line equal to the arc of a circle.

A square root is a number which multiplied into itself produced a number.

DEFINITION.

And the number so produced is called a square number. For example: Because 10 multiplied by 10 makes 100; the root is 10, and the square number 100.

CONSEQUENT.

In the natural row of numbers, as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, &c. every one is the square of some number in the same row. But square numbers (beginning at 1) intermit first two numbers, then four, then six, &c. So that none of the intermitted numbers is a square number, nor has any square root.

PROP. I.

A square root (speaking of quantity) is not a line, such as Euclid defines, without latitude, but a rectangle.

Suppose A B C D be the square, and A B, B C, C D, D A, be the sides, and every side divided into 10 equal parts, and lines drawn through the opposite points of division; there will then be made 100 lesser squares, which taken altogether are equal to the square A B C D. Therefore the whole square is 100, whereof one square is an unit; therefore 10 units, which is the root, is ten of the lesser squares, and consequently has latitude; and therefore it cannot be the side of a square, which, according to Euclid, is a line without latitude.

CONSEQUENT.

It follows hence, that whosoever taketh for a principle, that a side of a square is a mere line without latitude, and that the root of a square is such a line (as Dr. Wallis continually does) demonstrates nothing. But if a line be divided into what number of equal parts soever, so the line have breadth allowed it (as all lines must, if they be drawn), and the length be to the breadth as number to an unit; the side and the roof will be all of one length.

PROP. II.

Any number given is produced by the greatest root multiplied into itself, and into the remaining fraction. Let the number given be two hundred squares, the greatest root is 14(4)/(14) squares. I say that 200 is equal to the product of 14 into itself, together with 14 multiplied into (4)/(14). For 14 multiplied into itself makes 196. And 14 into (4)/(14) makes (56)/(14) which is equal to 4. And 4 added to 196 maketh 200; as was to be proved. Or take any other number 8, the greatest root is 2; which multiplied into itself is 4, and the remainder (2)/(4) multiplied into 2, is 4, and both together 8.

PROP. III.

But the same square calculated geometrically by the like parts, consisteth (by Euclid II. 4) of the same numeral great square 196, and of the two rectangles under the greatest side 14, and the remainder of the side, or (which is all one) of one rectangle under the greatest side, and double the remainder of the side; and further of the square of the less segment; which altogether make 200, and moreover (1)/(49) of those 200 squares, as by the operation itself appeareth thus:

The side of the greater segment is 14(4)/(14)
14(4)/(14)
Which multiplied into itself makes 200.

The product of 14, the greatest segment, into the two fractions (4)/(14), that is, into (4)/(14) (or into twice (2)/(14)) is (56)/(14) (that is 4); and that 4 added to 196 makes 200.

Lastly, the product of (2)/(14) into (2)/(14) or (1)/(7) into (1)/(7) is (1)/(49). And so the same square calculated by roots is less by (1)/(49) of one of those two hundred squares, than by the true and geometrical calculation; as was to be demonstrated.

CONSEQUENT.

It is hence manifest, that whosoever calculates the length of an arc or other line by the extraction of roots, must necessarily make it shorter than the truth, unless the square have a true root.

-------

_The Radius of a Circle is a Mean Proportion between the Arc of a
Quadrant and two-fifths of the same._

Describe a square A B C D, and in it a quadrant D C A. In the side D C take D T two-fifths of D C, and between D C and D T a mean proportional D R, and describe the quadrantal arcs R S, T V. I say the arc R S is equal to the straight line D C. For seeing the proportion of D C to D T is duplicate of the proportion of D C to D R, it will be also duplicate of the proportion of the arc C A to the arc R S, and likewise duplicate of the proportion of the arc R S to the arc T V.

Suppose some other arc, less or greater than the arc R S, to be equal to D C, as for example _r s_: then the proportion of the arc _r s_ to the straight line D T will be duplicate of the proportion of R S to T V, or D R to D T. Which is absurd; because D _r_ is by construction greater or less than D R. Therefore the arc R S is equal to the side D C, which was to be demonstrated.

COROL.

Hence it follows that D R is equal to two-fifths of the arc C A. For R S, T V, D T, being continually proportional, and the arc T V being described by D T, the arc R S will be described by a straight line equal to T V. But R S is described by the straight line D R. Therefore D R is equal to T V, that is to two-fifths of C A.

And your said servant most humbly prayeth you to consider, if the demonstration be true and evident, whether the way of objecting against it by square root, used by Dr. Wallis; and whether all his geometry, as being built upon it, and upon his supposition of an infinite number, be not false.

CONSIDERATIONS

UPON THE ANSWER OF DOCTOR WALLIS

TO THE

THREE PAPERS OF MR. HOBBES.

Dr. Wallis says, all that is affirmed, is but _if we_ SUPPOSE _that, this will follow_.

But it seemeth to me, that if the supposition be impossible, then that which follows will either be false, or at least undemonstrated.

First, this proposition being founded upon his _Arithmetica Infinitorum_, if there he affirm an absolute infiniteness, he must here also be understood to affirm the same. But in his thirty-ninth proposition he saith thus: “_Seeing that the number of terms increasing, the excess above sub-quadruple is perpetually diminished, so at last it becomes less than any proportion that can be assigned; if it proceed in infinitum it must utterly vanish. And therefore if there be propounded an infinite row of quantities in triplicate proportion of quantities arithmetically proportioned (that is, according to the row of cubical numbers) beginning from a point or 0; that row shall be to a row of as many, equal to the greater, as 1 to 4._” It is therefore manifest that he affirms, that in an infinite row of quantities the last is given; and he knows well enough that this is but a shift.

Secondly, he says, that usually in Euclid, and all after him, by _infinite_ is meant but, more than any assignable _finite_, or the greatest possible. I am content it be so interpreted. But then from thence he must demonstrate those his conclusions, which he hath not yet done. And when he shall have done it, not only the conclusions, but also the demonstration, will be the same with mine in Cap. XIV. Art. 2, 3, &c. of my book _De Corpore_. And so he steals what he once condemned. A fine quality.

Thirdly, he says, (by Euclid’s tenth proposition, but he tells not of what book), that a line may be bisected, and the halves of it may again be bisected, and so onwards infinitely; and that upon such supposed section infinitely continued, the parts must be supposed infinitely many.

I deny that; for Euclid, if he says a line may be divisible into parts perpetually divisible, he means that all the divisions, and all the parts arising from those divisions, are perpetually finite in number.

Fourthly, he says, that there may be supposed a row of quantities infinitely many, and continually increasing, whereof the last is given.

It is true, a man may say, (if that be supposing) that white is black: but, if _supposing_ be _thinking_, he cannot suppose an infinite row of quantities whereof the last is given. And if he say it, he can demonstrate nothing from it.

Fifthly, he says (for one absurdity begets another) _that a superficies or solid may be supposed so constituted as to be_ infinitely long, _but_ finitely great, _(the breadth continually decreasing in greater proportion than the length increaseth), and so as to have no centre of gravity. Such is Toricellio’s Solidum Hyperbolicum acutum, and others innumerable, discovered by Dr. Wallis, Monsieur Fermat, and others. But, to determine this, requires more of geometry and logic, (whatsoever it do of the Latin tongue), than Mr. Hobbes is master of._

I do not remember this of Toricellio, and I doubt Dr. Wallis does him wrong and Monsieur Fermat too. For, to understand this for sense, it is not required that a man should be a geometrician or a logician, but that he should be mad.

In the next place, he puts to me a question as absurd as his answers are to mine. Let him ask himself, saith he, if he be still of opinion, _that there is no argument in natural philosophy to prove that the world had a beginning_. First, whether, in case it had no beginning, there must not have passed an infinite number of years before Mr. Hobbes was born. Secondly, whether, at this time, there have not passed more, that is, more than that infinite number. Thirdly, whether, in that infinite (or more than infinite) number of years, there have not been a greater number of days and hours, and of which, hitherto, the last is given. Fourthly, whether, if this be an absurdity, we have not then, (contrary to what Mr. Hobbes would persuade us), an argument in nature to prove the world had a beginning.

To this I answer, not willingly, but in service to the truth, that, by the same argument, he might as well prove that God had a beginning. Thus, in case he had not, there must have passed an infinite length of time before Mr. Hobbes was born; but there hath passed at this day more than that infinite length, by eighty-four years. And this day, which is the last, is given. If this be an absurdity, have we not then an argument in nature to prove that God had a beginning? Thus it is when men entangle themselves in a dispute of that which they cannot comprehend. But, perhaps, he looks for a solution of his argument to prove that there is somewhat greater than infinite; which I shall do so far as to show it is not concluding. If from this day backwards to eternity be more than infinite, and from Mr. Hobbes his birth backwards to the same eternity be infinite, then take away from this day backwards to the time of Adam, which is more than from this day to Mr. Hobbes his birth, then that which remains backwards must be less than infinite. All this arguing of infinites is but the ambition of school-boys.

TO THE LATTER PART OF THE FIRST PAPER.

There is no doubt if we give what proportion we will of the radius to the arc, but that the arc upon that arc will have the same proportion. But that is nothing to my demonstration. He knows it, and wrongs the Royal Society in presuming they cannot find the impertinence of it.

My proof is this: that if the arc on T V, and the arc R S, and the straight line C D, be not equal, then the arc on T V, the arc on R S, and the arc on C A, cannot be proportional; which is manifest by supposing in D C a less than the said D C, but equal to R S, and another straight line, less than R S, equal to the arc on T V; and anybody may examine it by himself.

I have been asked by some that think themselves logicians, why I proceeded upon ⅖ rather than any other part of the radius. The reason I had for it was, that, long ago, some Arabians had determined, that a straight line, whose square is equal to 10 squares of half the radius, is equal to a quarter of the perimeter; but their demonstrations are lost. From that equality it follows, that the third proportional to the quadrant and radius, must be a mean proportional between the radius and ⅖ of the same. But, my answer to the logicians was, that, though I took any part of the radius to proceed on, and lighted on the truth by chance, the truth itself would appear by the absurdity arising from the denial of it. And this is it that Aristotle means, where he distinguishes between a direct demonstration and a demonstration leading to an absurdity. Hence it appears that Dr. Wallis’s objections to my _Rosetum_ are invalid as built upon roots.

TO THE SECOND PAPER.

First, he says that it concerns him no more than other men, which is true. I meant it against the whole herd of them who apply their algebra to geometry. Secondly, he says that a bare number cannot be the side of a square figure. I would know what he means by a bare number. Ten lines may be the side of a square figure. Is there any number so bare, as by it we are not to conceive or consider anything numbered? Or, by 10 nothings understands he bare 10? He struggles in vain, his conscience puzzles him. Thirdly, he says 10 squares is the root of 100 square squares. To which I answer, first, that there is no such figure as a square square. Secondly, that it follows hence, that a root is a superficies, for such is 10 squares. Lastly, he says that, neither the number 10, nor 10 soldiers, is the root of 100 soldiers; because 100 soldiers is not the product of 10 soldiers into 10 soldiers. This last I grant, because nothing but numbers can be multiplied into one another. A soldier cannot be multiplied by a soldier. But no more can a square figure by a square figure, though a square number may. Again, if a captain will place his 100 men in a square form, must he not take the root of 100 to make a rank or file? And are not those 10 men?

TO THE THIRD PAPER.

He objects nothing here, but that _the side of a square is not a superficies, but a line_, and that a _square root (speaking of quantity) is not a line, but a rectangle_, is a contradiction. The reader is to judge of that.

To his scoffings I say no more, but that they may be retorted in the same words, and are therefore childish.

And now I submit the whole to the Royal Society, with confidence that they will never engage themselves in the maintenance of these unintelligible doctrines of Dr. Wallis, that tend to the suppression of the sciences which they endeavour to advance.

LETTERS
AND OTHER PIECES.

LETTERS AND OTHER PIECES.

==========

I.
A LETTER FROM MR. HOBBS TO MY MR.[2]

HONORABLE SIR,

Though I may goe whither and when I will for anie necessity you have of my service, yet there is a necessity of good manners that obliges me as yo^r servant to lett you knowe att all times where to find me. Wee goe out of Paris 3 weekes hence, or sooner, towards Venice, but by what way I knowe not, because the ordinary high way through the territory of Milan is encumbered with the warre betweene the French and the Spaniards. Howsoever, wee have to be there in October next. If you require anie service that I can doe there, it may please you to convey your command by Devonshire house. But if you command me nothing, I have forbidden my letters to look for answer: their busines being only to informe and to lett you knowe that the image of your noblenes decayes not in my memory, but abides fresh to keepe me eternally

Your
THO. HOBBS.

-----

Footnote 2:

This letter is to be found in the British Museum, amongst the
Lansdowne MSS. 238, entitled “a collection of letters to and from
persons of eminence in the reigns of Elizabeth, James I, and Charles
I, made by some person in the service of Sir Gervas Clifton”. It is
without date: but the allusion to the war between France and Spain,
and the passage in the VITA THO. HOBBES, “Anno sequente qui erat
Christi 1629, rogatus a nobilissimo viro domino Gervasio Clifton”, &c.
(p. xiv), show that it must have been written in either 1629 or 1630.

-----

II.
TO A FRIEND IN ENGLAND.

WORTHY SIR,

I have been behind hand with you a long time for a letter I received of yours at Angers, that place affording nothing wherewith to pay a debt of that kind, all matter of news being sooner known in England than here: and the news you writ me was of that kind, that none from England could be more welcome, because it concerned the honour of Welbeck and Clifton, two houses in which I am very much obliged.

Monsieur having given the slip to the Spaniards at Bruxelles, came to the King about ten days ago at St. Germains, where he was received with great joy. The next day the Cardinal entertained him at Ruelle: and the day after that he went to Limours, where he is now, and from thence he goes away shortly to Bloys, to stay there this winter. The Cardinal of Lyons is going to Rome to treat about the annulling of Monsieur’s marriage, which is here by Parliament declared void, but yet they require the sentence of the Pope. There goes somebody thither on the part of his wife, to get the marriage approved: but who that is, I ’know not. The Swedish party in Germany is in low estate, but the French prepare a great army for those parts, pretending to defend the places which the Swedes have put into the King of France his protection, whereof Philipsbourgh is one; a place of importance for the Lower Palatinate. This is all the French news.

For your question, _why a man remembers less his own face, which he sees often in a glass, than the face of a friend that he has not seen of a great time_, my opinion in general is, that a man remembers best those faces whereof he has had the greatest impressions, and that the impressions are the greater for the oftener seeing them, and the longer staying upon the sight of them. Now you know men look upon their own faces but for short fits, but upon their friends’ faces long time together, whilst they discourse or converse together; so that a man may receive a greater impression from his friend’s face in a day, than from his own in a year; and according to this impression, the image will be fresher in his mind. Besides, the sight of one’s friend’s face two hours together, is of greater force to imprint the image of it, than the same quantity of time by intermissions. For the intermissions do easily deface that which is but lightly imprinted. In general, I think that lasteth longer in the memory which hath been stronglier received by the sense.

This is my opinion of the question you propounded in your letter. Other new truths I have none, at least they appear not new to me. Therefore if this resolution of your first question seems probable, you may propound another, wherein I will endeavour to satisfy you, as also in any thing of any other nature you shall command me, to my utmost power; taking it for an honour to be esteemed by you, as I am in effect,

Your humble and faithful servant
THO. HOBBES.

_Paris, Oct. 21/31, 1634._

My Lord Fielding and his Lady came to Paris on Saturday night last.

III.
TO MY WORTHY FRIEND MR. GLEN.[3]

WORTHY SIR,

I received here in Florence, two days since, a letter from you of the 19th of January. It was long by the way; but when it came it did thoroughly recompence that delay. For it was worth all the pacquets I had received a great while together. All that passeth in these parts is equally news, and therefore no news; else I would labour to requite your letter in that point, though in the handsome setting down of it, I should still be your inferior.

I long infinitely to see those books of the Sabbaoth[4], and am of your mind they will put such thoughts into the heads of vulgar people, as will confer little to their good life. For when they see one of the ten commandments to be _jus humanum_ merely, (as it must be if the Church can alter it), they will hope also that the other nine may be so too. For every man hitherto did believe that the ten commandments were the moral, that is, the eternal law.

I desire also to see Selden’s _Mare Clausum_, having already a great opinion of it.

You may perhaps, by some that go to Paris, send me those of the Sabbaoth, for the other being in Latin, I doubt not to find it in the Rue St. Jaques.

We are now come hither from Rome, and hope to be in Paris by the end of June. I thank you for your letter, and desire you to believe that I can never grow strange to one, the goodness of whose acquaintance I have found by so much experience. But I have to write to so many, that I write to you seldomer than I desire; which I pray pardon, and esteem me

Your most affectionate friend
and humble servant
THO. HOBBES.

_Florence, Apr. (6)/(16) 1636._

My Lord and Mr. Nicholls, and all our company commend them to you.

-----

Footnote 3: