A Fit Conjucture of Religion and Learning, Book 2 · Chapter 4
CHAP. IV. Of the Mathematicks.
THe Mathematicks are so named, because they are not learned without a teacher.Ma…he…is à foreign disco, quod hanc prae reliquis pueri olim discebant. Explorandae eorum indolis & ingenii ergo. Ames. Technomet. Mathemata quod sine praece▪ tore percipi nequeant, dicta sunt. Exinde scient… Mathematicae di…imitus solae discerentur, antequam animum adpellerent ad Philosophiam natur…el civile… Mathemat. Mathematicas disciplinas omnium accuratissimas & utilissimas multis …Plato in …h…laeo. Ideoque primas has artes apud Graecos, tanquam doctrinae quaedam rudimenta discebant …cer ad reliquas scientias aditum aperiunt; Nam & ingenia acuunt eorumque solertiam augent, ad retum▪ co…ant, & celeritatem quandam percipiendi conciliant. Et quoniam imitatione percipiuntur, aprae puerilibus animis qu… ea potissimum ducuntur. Ioan. à Wovver. De Polymathia. Tractat. cap. 23. Non solum in …m…h strationum certitudine & scientiae pulchritudine: sed & veritate multiplicique cognitionis copia reliqua artium genera longe an… disciplinae, adeò ut non immerito ob id Mathematicae, quasi solae disciplinarum nomine, ob excellentiam, dignae à veteribus appellatae sint. Tych. Brah. Orat. de Discipl. Mathem. Vide Ioannis Pellii Orat. inaugural.
When Alexander bad his Mathematick master make his lessons easier to him then in his ordinary way to others, he answered him well, The Mathematicks were alike difficult to all.
The ancient Philosophers taught their Pupils Mathematicks very soon, as appears by Aristotle and others, they confirming their Propositions by mathematical demonstrations, the knowledge whereof must be presupposed. See Tych. Brah. Orat. De Discip. Mat.
The Mathematicks are a most accurate and profitable study for the evidence and certainty of their demonstrations.
Abraham was famous for teaching the Mathematicks first among the Chaldeans, and after he taught the same A… as Historians write.
The old Mathematicians drew their proportions in dust with a Compass, the better to shew what they would. Tully cals it learned dust.De natura Deorum. Pers. Sat. 1.
Et secto in pulvere metas.
Archimedes was doing this when Siracuse was taken.
Princes and great persons have studied the Mathematicks, and those that studyViri nobiles, ac principes, qui non lucrandi, sed Philosophandi causa literis dant operam, Mathematicis maximè studiis delectantur. Inter quos celeberrimi extiterunt, ex antiquis quidem Archimedes Regum Siciliae consanguineus; Boetius vir consularis; Alphonsus Rex Hispaniarum: nostra vero aetate Marchio Guidobaldus, Princeps Ticho Brahe; Franciscus Candalla, & alii complures, quorum monumenta in omne aevum permansura mundus mirabitur. Blancani Dissertat. De Natura Mathematicarum, cap. 3. them to purpose, had need to have some Princes or great persons to encourage them therein by their bounty, in respect of the chargeablenesse of their instruments, as Tycho Brahe somewhere saith.
Mathematicks are either Pure or Mixt:
To pure Mathematicks those Sciences are referred, which handle QuantityQui Mathematicus audire cupit, non Arithmeticam modo & Geometriam▪ quae Mathesin propriè dictam & ab omni materia abstractam constituunt, verum & Astronomiam Geographiam, Opticam, Mechanicam, & Staticam, Musicam, Geodaesiam & Architecturam militarem, probè calleat oportet. Hortenfii Dissertat. De studio Mathematico recte instituendo. altogether abstracted from matter; and Physical axioms, they are two, Geometry and Arithmetique, the one handling continued Quantity, the other dissevered Quantity, or number.
To mixt Mathematicks Astronomy and Musick are usually referred, and the Opticks.
I. Arithmetique is an Art of numbring well.Arithmetica est doctrina bene numerandi. Rami. Arichm. l. 1. Arithmetica & Musica separatim magnitudinem, id est, numeros considerat. Geometria magnitudinem conjunctam, stabilem & immotam, Astronomia mobilem. Arithmetica merito prima, quae reliquis omnibus usui: ab ea & musica principia sumit. Nam ut Arithmetica proprieta…es numerorum absolute considerat: ita musica easdem in so…orum differentiis & proportionibus contemplatur. Ratio enim harmoniae quae est musicae, numeris & figuris constat. Harum auxiliis Geometria & Astronomia utuntur. Nam sine Arithmetica, Geometria subsistere non potest. Numerorum enim cognitio prior denotatione linearum. Et ex numeris lineae inventae, ex lineis figurae productae. Astronomia à Geometria lineas & figuras ad coelorum orbes, siderum conversiones considerandas, corporum coelestium intervalla colligenda, eorumque magnitudinem describendam adsumit. Ioan▪ à Wovver. De Polymath. Tractat. c. 23. Arithmetica subtilior est quam Geometria. Quodcunque enim vel accipit vel probat Geometria per numeros dirigi potest ab iis enim pendet …o…a magnitudinum consideratio Mathematicarum. At infinita procedent in numeris, quorum multa non potuerunt in magnitudinum natura comprehendi. Scalig. De Subtil▪ ad Cardan. Exercit. 331. Denique si quid est, in quantitate subtilitatis, id omne ex numerorum ratione haustum est. Pender igitur Geometriae dispositio ab Arithmetici constitutione. Quae res apud neminom controversa est. Subtilior igitur haec, quae illius statuit principia, ac fundamenta. Id. ibid.
Pythagoras first brought it into an Art, thence the Pythagoreans did swear per numerum quaternarium, quò nihil apud eos videtur esse perfectius. For there are four Elements, Fire, Air, Water, Earth; four times of the year, Spring, Autumn, Summer, Winter; four qualities of all things▪ Heat, Cold, Moisture, Drinesse, four Coasts of the Heaven▪ East, West, North, South.
It is the first of the Mathematicks, and usefull to all the rest.
One should get some skill in the vulgar▪ Arithmetique before he learn Geometry, and that either by a teacher, or by plain books, such as the Arithmetique of Gemma Frìfius, or Ramus with Snellius his Notes.
The principle of number is unity, as of magnitude punctum.
By this the Pythagoreans in times past covered the mysteries of all nature, and even of God himself, as if all things were agreeable to numbers.
The Phenicians write as all the Eastern Nations, from the right hand to the left: the reason why the outermost figure to the right hand in Arithmetique stands in the first place, they also being the inventers of that Science. George Sandys upon the 3d Book of Ovids Metam.Quàm mirabile illud veluti scientiarum monstrum, ac portentum, quod Algebram vocant? Nihil fortasse in tota peritiae Eucycopledia subtilius, profundius nihil, non humano ingenio par est, sed quid coelitus revelatum dixeris: numeros illos, quos surdos vocant, & qui nullo modo exprimi possont addit, subtrahit, multiplicat, dividit, perinde ac si numeri communes essent. Ii qui hanc callent eruditionem nihil in numerorum infinita ditione obscurum nihil arduum timeant. Blancani Dissertatio De natura Mathematicarum, cap. 4.
Some learned men speak much of the Algebra, as a famous invention in Arithmetique.
II. Geometry is an Art of measuring well.
The measures of things are taken from the parts of man, his finger, palm, foot,Geometria est ars bene metiendi, Rami Geomet. lib. 1. cap. 16. Geometria pars est Mathematicae scientiae, imò basis ac fundamentum, atque ut Philo scite loquitur, Metropolis omnium artium Mathematicarum. Nam ex ea, tanquam matrice civitate, derivatae sunt illae nobilissimae Coloniae, Optica, Astronomiae, caeterae; ipsaque adeò Arithmetica, cujus subjectum est simplicissimum, & Geometrico simplicius, remotiúsque à materia, utpote carens situ, Geometriae adminiculo stat, câque nititur, quod admirabile cuipiam videatur, ad conclusiones suas demonstrandas. Etsi ex altera parte non negarim Geometriam quoque in irrationalium praesertim doctrina, lucem & praesidium ab Arithmetica mutuari. Savil▪ Lect. 2. in Euclyd. Habet magnas utilitates Geometria, sed voluptates etiam partibus infinitis majores habet, quibusque fimile nihil in alia ulla disciplina reperie…u…: neque enim Grammaticus rhetor logicus, alioqui ettam▪ physicus, & Ethicus invenit in arte sua novi laetitia elatus, immolasse diis memoratur. Thales Milesius cum trianguli in circulo inscriptionem invenisset bovem diis immolavit. Ram. Orat. De sua professione. Vide plura ibid. cubit.
A barly corn is the least measure; the finger is the breadth of four barly corns; the palm, four fingers; the foot four palms. A cubit from the elbow to the end of the longest finger is a foot and an half.
From those measures ariseth the measure of a journey, a pace, a furlong; a mile, a geometrical pace is five foot, a furlong 125 paces, a mile a thousand paces, eight furlongs.
Quatuor ex granis digitus componitur unus,
Est quater in palmo digitus, quater in pede palmus,
Quinque pedes passum faciunt, passus quoque centum
Viginti quinque stadium da…t, sed miliare
Octo dabunt stadia, duplatum dat tibi Leuca.
The Romans used a foot for their measure, as the Jews did a cubit.
Plato wrote in the door of his study, foreign, Let no man ignorant of Geometry enter in here.
Sir Henry Savill in his ninth Lecture upon Euclyde, saith, Ars est omnium sine controversia certissima, it is without controversie the most certain Art of all.
Tyc. Brahe in his Oration, De Disciplinis Mathematicis, shews, That divers Arts flow from Geometry, Astronomy, Opticks; that of dialing, Geography, Architecture, and the Mechanicks,
All with one consent ascribe the invention of Geometry to the Egyptians. ForGeometriae olim in Aegypto & Graecia plurimi certatim incubuerunt Artifices: quorum singuli ad artis constitutionem aliquid contulerunt. Euclides Megarensis uno complexus volumine, artis elementa continuo ordine & magna solertia ita tradidit, ut à quovis mediocris ingenii acumine praedito non difficulter percipi possent. …ych. Brah. Orat. De Discip. Math. Quaestio de quadratura circuli multa praeclara exercuit ingenia. Num quaestionis subtilitatem an ingenii humani imbecillitatem, in causa esse dicemus, quod seculis tot fuere desudatum in nodo solvendo, qui adhuc sit involutus. Vossius de Scient. Mathem. lib. 1. cap. 16. when the limits of their possessions were confounded by reason of the annual overflowing of Nilus, to restore these, the reason and way of measuring fields was invented, and so Geometry may seem to have drawn its first original from the Husbandmens measuring their grounds. Thales first brought Geometry out of Egypt into Greece; Pythagoras amplified it. To whom many succeeded; Euclide hath brought the Art to that perfection, that he hath cut off hope from all his posterity of enlarging it.
The great difficulty among Geometricians is about the quadrature of a circle, which some say, is not faisible.
III. Musick. This is either Vocal or Instrumental; Some preferre theTherpander Thebanus Musicus celeberrimus primus de musica scripsit. Bla…ca…i. Chronol. Clarorum Mathematicorum. Musicae principia primum Pythagoras ad artem reduxit. Geneb Chron. l. 2. Hic primus dicitur ex melleorum ictibus vocum diversa…um numeros, modes, inflexionesque perpendendo, Musicam artem divina quadam industria confecisse. Cytharam tradunt invenisse Orpheum, Fistulam Pan Arcadius primus inflavit. Ringelbergius De Ratione Studii. Quomodo different foreign (quae tria Hebraeor…m instrumenta musica praecipua erant, & junguntur Nehemiae, Cap. xii.) Iosephus optimè dicet, lib. 7. Antiq. cap. x. & è nostris Basilius, Chrysostomus, Hieronymus. Vossius. De vitiis Sermonis. l. 4 c. 14. Vocall.
Mercer on the 4th of Genesis ver. 21. saith, Iubal was the inventer of Instrumental Musick.
Pythagoras Musicae theoricam ex Fabri malleis adinvenit. Blancani Chronologia Clarorum Mathemat. Pythagoras invented the Art of Musick from the Smiths hammers.
One seeing another playing excellently upon the Lute, said, He thought his soul was in his fingers.
Leur esprit…s' ensuit au bout des doigts, saith Du Bartas. 1. Sept.
Pythagoras, Aristoxenus, Nicomachus, Philolaus, Alypius, Ptolomaus handled Musick professedly, Boetius, Martianus Capella, and venerable Bede, and Mersennus have written also of Musick.
IV. Astronomy.Astronomia dicitur foreign, quia docet, qua lege moveantur ac reguntur astra. Vossius de Scient. Math. c. 28. Admiranda prorsus & divina Astronomia, subjecti nobilita…e cunctis prior, apta & necessaria omnibus in quibus ulla temporis ratio. Ioan à Wovver. De Polym. Tract. c. 25. In Astronomicis à doctrina Sphaerica initium faciat, legendo Institutiones Astronomicas Peuceri, Metii, Maestlini, Crugeri, & Commentaria Clavii in Sphaeram. Ioh. de Sacrobosco. Hortensii Dissertat. do studio Mathemat. recte instituendo.
The word signifies a Doctrine of the Laws or Rules of the measure and motion of the Stars.
The Subject of this Science is the Heavens with the Stars. Diodorus, Pliny and others say, that Atlas was therefore thought to sustain heaven with his shoulders, because he made a Sphere in which the whole heaven was represented.
It exceeds not only Arithmetique and Geometry, but all other Arts in this kinde. Nam & antiquitate & nobilitate ac cognitionis sublimita…e prae reliquis omnibus maximam sibi vendicat laudem. Tych. Brah▪ Orat. De Discip. Mathem.
Peter Ramus in proaem. Mathem. complained, because we have not now Astronomy free from Hypotheses, and therefore easie, as the Egyptians and Babylonians had. But saith Gassendus (Praefat. In Tychonis Brahaei vitam) though some Hypotheses be more simple, and so more easie then others, yet Astronomy cannot consist without some.
Quod autem celeberrimus ille nostri aevi Philosophus Petrus Ramus existimarit, sine Hypothesibus per Logicas rationes Astronomiam constitui posse, caret fundamento, &c. Tycho Brahe Epist. Astronom. lib. 1. Christoph. Rothman. Vide plura ibid.
Thales Milesius was the first who studied the Stars, Gassendus (in the Preface before-quoted,Quantum verò ad nos in ast…orum scientia pervenit, Hipparchus Rhodius, primus ejus parens censetur. Is enim fixarum loca diligenter observavit. Solisque & Lunae cursus per plurimos subsequentes annos designavit. Ac in caeterorum Planetarum …o…iusque Coeli observationibus diligentissimus fuit. Hujus inventis & considerationibus Ptolomaeus Alexandrinus cum suis collatis, totam artem in certas Hypotheses, unaque in numeros redegit, & immortalem apud omnem posteritatem hinc merito sibi p…perit memoriam. Nostra vero ae…ate Nicolaus Copernicus, quem alterum Ptolomaeum non immeritò dixeris, admiranda ingenii solertia Hypothesibus aliter constitutis coelestium motum, scientiam ita restauravit, ut nemo ante ipsum exactiùs de syderum motibus sit philosophatus. Tych. Brah. O…at. De Discip. Mathem. Ex his duobus artificibus Ptolomaeo & Copernico, omnia illa quae nostra ae…ate, in astrorum revolutionibus, perspects & cognita habemus, constituta ac tradita sunt. Id. ibid. and their lives) mentions Eudoxus, Hipparchus, Theon, Pappus, Alphraganus, Alphonsus King of Castile, Tycho Brahe, Copernicus, Peurbachius, and Regio-Montanus for famous Astronomers, Blanchinus, and Ptolomy, to which Gassendus himself (the Professour of Astronomy now at Paris) may deservedly be added.
V. Opticks. This is a Greek word. It is called Perspectivè in Latine, a barbarousOptices dignitatem cognoscimus ex nobilitate organi ejus, quod sunt oculi, praecipium cognitionis instrumentum: his enim omnem Philosophiam debemus. Etiam Opticae dignitatem ostendit objecti praestantia; Cùm luce & coloribus, nihil sit melius vel jucandius. Vossius de scientiis Mat. c. 24 Unam Opticen video solam esse artem, quae caeterarum communis splendor dici possit, quae solis instar, reliquis lucem impertiat, rerum in naturae majestate abditarum nubem dispellat. Pena de usu optl…es. Vide Volat. Comment. Urban. l. 35. but significant word, Optica est ars bene videndi. The Opticks is an Art of seeing well.
Light and colour are visible by themselves, so it is said; Light indeed is, but colours will not be visible in the dark.
We should examine and weigh the writings of the chief Astronomers by this Lesbian Rule.
Luceus was the first Inventer of the metal-Mines, and hence arose the common fable, that he also saw those things which were under the earth.
Pisanus was the Author of the common Perspective.
Alhazeus and Vitellio, and Pena on Euclide, and Peckam Archbishop of Canterbury have done best on the Opticks.
Galen skilfull in the Opticks as well as Physicks, gives this demonstration, why one and the same appears one to two eyes, Eadem est basis pyramidum sub radiis ab oculo missis comprehensarum.
There are perfect demonstrations in the Opticks; as why a hollow glasse burns, because the Sun-beams reflected are there gathered together; why a staff appears broken in the water, because it is seen by lines refracted, through a double medium of air and water.