Plus ultra, or, The progress and advancement of knowledge since the days of Aristotle in an account of some of the most remarkable late improvements of practical, useful learning, to encourage philosophical endeavours : occasioned by a conference with one of the notional way / by Jos. Glanvill.

CHAP. III. Another great Advantage of late Times, from the Improvements of Mathe∣maticks; particularly of Arith∣metick, Algebra, and Geometry, discourst by Instances. (Book 3)

I Proceed now to my THIRD Instance of ARTS (if the Reverend Logi∣cian will give me leave to use the word in this large sense) which are Advantages for deep search into Nature, and have been con∣siderably advanc'd by the Industry and culture of late Times, above their an∣cient

Stature. And the Instance was,

(III.) THE MATHEMATICKS. That these are mighty helps to practical and useful Knowledge, will be easily confest by all, that have not so much ignorance as to render them incapable of information in these matters. And the Learned Ge∣rand Vossius hath proved it by induction in particulars. And yet it must be acknow∣ledged that Aristotle, and the disputing Phi∣losophers of his School, were not much ad∣dicted to those noble Inquisitions: For Pro∣clus the Commentator upon Euclide, though he gives a very particular Catalogue of the Elder Mathematicians, yet hath not menti∣oned Aristotle in that number. And though Diogenes Laertius takes notice of a Book he inscribed 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, another, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, and a Third, yet extant, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉; Yet it appears not that these were things of very great value; and Aristotle's Metaphy∣sical procedure, even in Physical Theories, the, genius and humour of his Principles, and the aiery contentions of his Sect, are huge pre∣sumptions that this Philosopher was not very Mathematical. And his numerous suc∣ceeding Followers, were certainly very lit∣tle conversant in those generous Studies. I have elsewhere taken notice, that there is

more publish'd by those Disputing men on some paultry trifling Question about ens Rationis, and their Materia prima, than hath been written by their whole number upon all the vast and useful parts of Ma∣thematicks and Mechanicks. There was a time when these were counted Coniurati∣ons; and I do not very well know the reason of the Reverend Disputers dis∣pleasure at my Discourse about Diop∣trick Tubes (of which you will hear in the process) except he was under the dread of some such phansie, and believed there was Magick in Opticks. It would require much skill in those Sciences, to draw up the full History of their Advancements; I hear a very accurate Mathematician is upon it: And yet to fill up my Method, I'le adventure at some imperfect Suggestions about the Inventions and Improvements of this kind: And I begin,

(I.) With ARITHMETICK, which is the handmaid to all the other parts of Mathematicks. This indeed Pythagoras is said to have brought from the Phoenicians to the Graecians: but we hear no great mat∣ter of it till the days of Euclide: not the Euclide that was the Contemporary of Plato,

and Hearer of Socrates; but the famed Mathematician of that Name, who was af∣ter Aristotle, and at 90 years distance from the former. This is the first Person among the Ancients, that is recorded by the exact Vossius to have done any thing accurately in that Science. After him it was advanced by Diophantus, methodized by Psellus, illustra∣ted among the Latins by L. Apuleius, and in later times much promoted by Car∣dan, Gemma Frisius, Ramus, Clavius, and divers more modern Artists, among whom I more especially take notice of that Inge∣nious Scot the Lord Napier,

Who invented the Logarithmes, which is a way of computing by artificial Numbers, and avoiding the taedium of Multiplication and Division. For by this Method all those Operations are performed by Addition and Substraction, which in natural Numbers were to be done those longer ways. This Invention is of great use in Astronomical Calculations, and it may be applied also to other Accompts. Besides this, the same Learned Lord found an easie, certain, and compendious way of Account∣ing by Sticks, called Rabdology; as also Computation by Napier's Bones: Both these

have been brought to greater perfection by others, since their first Discovery; particu∣larly by Vrsinus and Kepler.

To them I add the Decimal Arithmetick, which avoids the tedious way of computing by Vulgar Fractions in ordinary Accompts, and Sexagenaries in Astronomy; exceed∣ingly and lately improved by our famous Oughtred, and Dr. Wallis a Member of the ROYAL SOCIETY. (If I should here subjoyn the Helps this Art hath had from the Works and Endeavours of Ana∣tolius, Barlaam, Maximus Palanudes, Ne∣morarius, Florentinus Bredonus, Pisanus, Orentius; and in this Age, from those of Adrianus Romanus, Henischius, Cataldus, Malapartius, Keplerus, Briggius, Crugerus, and a vast number reckon'd up by Vossius, I should be tedious on this Head; and therefore I pass lightly over it, and pro∣ceed,

(II.) To ALGEBRA, of universal use in all the Mathematical Sciences, in Com∣mon Accompts, in Astronomy, in taking Di∣stances and Altitudes, in measuring plain and solid Bodies, and other useful Operati∣ons. The first noted Author in this Me∣thod was Diophantus, who lived long since the Idol of Disputers. He, and those other

Ancients that used it, performed their Al∣gebraical Operations by Signs and Characters suted to the several Numbers, and powers of Numbers, which they had occasion to use in solving Problems: But the later Ma∣thematicians have found a far more neat and easie way, viz. by the Letters of the Al∣phabet, by which we can solve many Pro∣blemes that were too hard for the Ancients, as far as can be discovered by any of their remaining Works. For there were many affected Aequations (as they call them) that did not equally ascend in the Scale of Pow∣ers, that could not be solv'd by the elder Methods; whereas the acute Vieta, a Ma∣thematician of this last Age, affirms, he could resolve any Probleme by his own Im∣provements. Besides him, our excellent Oughtred another, lately mentioned, did much in this way. But the inimitable Des Cartes hath vastly out-done both former and later Times, and carried Alge∣bra to that height, that some considering men think Humane Wit cannot advance it further. I will not say so much: but no doubt he hath performed in it things deser∣ving vast acknowledgment, of which you shall hear more anon. And from hence I step,

(III.) To the Consideration of GEO∣METRY, which is so fundamentally use∣ful a Science, that without it we cannot in any good degree understand the Arti∣fice of the Omniscient Architect in the composure of the great World, and our selves. 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, was the excellent saying of Plato; and the Vni∣verse must be known by the Art whereby it was made. So that what Galileo notes of Aristotle, is a great sign of his defects, viz. That he reprehended his Venerable Ma∣ster for his Geometrical Sublimities, accusing him that he receded from the solid Methods of Philosophizing, through his too much in∣dulging that Study; Which is so far from being likely, that Geometry is little less than absolutely necessary to solid and real Philo∣sophy; and as I intimated, 'tis next to im∣possible to be a judicious and accurate Phi∣losopher, and want it. Upon which ac∣count, Plato admitted none to his School, but those that were acquainted with that Science: Which practice the mentioned excellent Modern, notes to be directly op∣posite to the Peripatetick genius; and some he knew great men of that way, dehorted their Disciples from it; which he intro∣duceth

one applauding as a wise Counsel, since Geometry would detect and shame the futilities of that National way. But not to take too large a compass, this is certain, That Geometry is a most useful and proper help in the affairs of Philophy and Life. 'Tis almost as clear from those former intimati∣ons, that Aristotle was not much enclined that way; and we know that his late Secta∣tors, have very seldome applied themselves to Geometrical Disquisitions.

The Result of which is, We must ex∣pect the Advantages of this Science, from the declining of his and their Empire; and I need not say expect it, they are both in pre∣sent view. And if after this you require accounts of the Improvements Geometry hath received, since the foundation of that Tyranny by the Man of STAGYRA, I shall offer you the best I have; and though I am conscious that they will be scant and defective, yet I hope sufficient for my pre∣sent purpose.

I note then from the celebrated Vossius, That Euclide was the first that brought Geo∣metry into a Method, and more accurately demonstrated those Principles, which be∣fore were scattered among the Greeks and Aegyptians, and not so cogently or carefully

proved. And Proclus reckons this Famous man as the Compiler and Demonstrator, not as the Inventor of the Elements; and two of these Books (viz. 14. & 15.) are ascri∣bed to Apollonius Pergaeus, who was his near∣est Successor in Fame for Mathematical Abili∣ties. This Geometrician improved the Sci∣ence by four Books of Conicks, publish'd of old; and three more have been lately (in the year 1661.) translated out of an Ara∣bick Manuscript in the Duke of Tuscany's Library, and are now abroad. This Ma∣nuscript Iacob Golius procured out of the East. Besides which, this Magnus Geome∣tra, as he was called, illustrated Euclide by his Learned Commentary upon him. But Archimedes of Syracuse, was a Person of the greatest renown for Geometrical and Mecha∣nical Performances; concerning which, Po∣lybius, Valerius, Plutarch, Livy, and others, have recorded prodigious things. This great Wit carried Geometry from general and idle Speculation, to the use and benefit of Man∣kind; whereas before him it was an anci∣ent and perverse Opinion, That this Know∣ledge ought not to be brought down to vulgar Service, but kept up in abstractive Contemplations: upon which score Archy∣tas and Eudoxus, those great Geometricians

before Euclide, were scared from the Mecha∣nical and Organical Methods, to the great hindrance of beneficial Improvements in that way. But the excellent Syracusian under∣stood, that this Science is not debased, but promoted and advanced by such Accommoda∣tions; and evinc'd the usefulness and excel∣lency of Geometry, in his admirable Paradox proposed before King Hieron (Datis viribus datum pondus tollere) [〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉] This Mathematician flourish'd 160 years after the time of Aristotle, who hath the name of the most ancient that writ in Mechanicks, though that Book of his be not mentioned either by Archimedes, Athenaeus, Hero, or Pappus, Mechanical Authors; and Cardan and Patricius affirm that Work to be none of Aristotle's: Whose ever it was, the Performance hath praise from the Learned, as explaining the gene∣ral Causes of Mechanical Geometry. But Archimedes was more practical and parti∣cular: And though Plutarch in the Life of Marcellus affirms he writ nothing; yet the contrary is abundantly proved by Gerard Vossius, who hath shewn that the Books extant under his Name, that contain so many great Maxims of Mechanicks, are genuine; and both Strabo and Pappus men∣tion

them as his. The Design of Archime∣des, of combining Mechanism and Geo∣metrick Theory, was after happily promo∣ted by Hero the Elder of Alexandria, who invented those ingenuous Automata, that move by Air and Wyres; concerning which he writ a Book that was Translated by Fre∣dericus Commandinus, as also he did ano∣ther De Machinis Bellicis, by which he well improved Geometrick Mechanicks: And Pappus particularly celebrates his exactness in solving the Deliaick Problem, De Cubo du∣plicando, acknowledging that he took most of his own Accounts about that matter, from that exquisite Man. Next him, I menti∣on Theodosius of Tripoli, who very much im∣proved Geometry by his three Books De Fi∣gur a Sphaerica, which afforded great assi∣stance to Ptolomy, Pappus, Proclus, and The∣on, in their Mathematical Endeavours. Me∣nelaus also, who lived in Trajan's time, contributed very much to the perfecting the Doctrine of Sphaericks, as Vitellio well knew, who was famous for those things which he borrowed from that Author. The Performances also of Ctesibius, who lived in the time of Ptolomaeus Physcon, are much celebrated by Pliny. He invented many things in Hydraulicks, and according to

Athenaeus, he was the first Contriver of Mu∣sical Organs. These were Mechanical: but Geminus Rhodius the Master of Proclus Ly∣cius, applied Logick to Geometry, out of par∣ticular Elements abstracting Vniversals. He demonstrated, That there are only Three similar Species of all Lines, viz. Right, Cir∣cular, and Cylindrical: And Perseus follow∣ing his steps, enrich'd Geometry with the In∣vention of three kinds of Crooked Lines, the Parabole, Hyperbole, and Elipsis; for which he express'd his extatick joy, as Thales, Pythagoras, and Archimedes did upon like occasions, in a Sacrifice to the Gods. But to be briefer, Pappus improved the Sphoe∣ricks; Theon more methodically digested the Elements of Euclide; Serenus Antinsensis discover'd, that the Section of a right Cy∣lindre, is the same with the Elipsis of a right Cone; Copernicus improved the Doctrine of Triangles; Ramus corrected and supplied Euclide, where his Principles were defe∣ctive; Maurolicus writ first of Secant Lines; Clavius much illustrated and pro∣moted the Doctrine of Tangents, Secants, Triangles, Right Lines, and Sphaericks, be∣sides what he did in his Comment upon Eu∣clide. I might mention with These, the worthy Performances of Cusanus, Pitiscus,

Snellius, Ambrosius Rhodius, Kepler, Francis∣cus à Schoten, and others, who contributed very eminently to the perfections and ad∣vancements of Geometry, and were late men.

But none have done in it like the excellent Persons whom I reserve for my last menti∣on; The chief are Vieta, Des Car∣tes, and Dr. Wallis.