# Philosophical Transactions of the Royal Society (1665-1886)/Volume 3/Number 38

Contents:
* Contents
* A Contrivance to make the Picture of any thing appear on a Wall, Cup-board, or within a Picture-frame, &c. in the midst of a Light room in the Day-times; or in the Night-time in any room that is inlightned with a considerable number of Candles.
* An Extract of a Letter, written from Leiden by Mr. Sam. Colepress to the Publisher, about the making of counterfeited Opal, and the Recovery of the Art making Red-glass.
* Some Animadversions written in a Letter by Dr John Wallis to a printed Paper entituled, Responsio Francisci Du Laurens ad Epistolam D. Wallisii ad Cl. Oldenburgium scriptam.
* An Account of two Books.
** I. REGNERI de GRAEF Med. D. De VIRORUM ORGANIS GENERATIONI INSERVIENTIBUS.
** II. LOGARITHMOTECHNIA NICOLAI MERCATORIS: discoursed of in a letter written by Dr. J. Wallis to the Lord Vis-count Brouncker; at whose desire the same Doctor in another Letter demonstrates his Method of Finding the Summ of Logarithmes. To which is added Mr. Mercatois illustration of his said Logarithmotechnia.
PHILOSOPHICAL
TRANSACTIONS.
Monday, August 17. 1668.
The Contents.
A Contrivance
THis Optical Experiment, here to be described, is New, though easy and obvious; and hath not, that I know, been ever made by any other person this way. It produces Effects not ​onely very delightful, but to such as know not the contrivance, very wonderful; so that Spectators, not well versed in Opticks, that should see the various Apparitions and Disappearances, the Motions, Changes, and Actions, that may this way be represented, would readily believe them to be super-natural and miraculous, and would as easily be affected with all those passions of Love, Fear, Reverence, Honour, and Astonishment, that are the natural consequences of such belief. And had the Heathen Priests of old been acquainted with it, their Oracles and Temples would have been much more famous for the Miracles of their Imaginary Deities. For by such an Art as this, what could they not have represented in their Temples? Apparitions of Angels, or Devils, Inscriptions and Oracles on Walls; the Prospect of Countryes, Cities, Houses, Navies, Armies; the Actions and Motions of Men, Beasts, Birds, &c. the vanishing of them in a cloud, and their appearing no more after the cloud is vanisht: And indeed almost any thing, that may be seen, may by this contrivance be very vividly and distinctly represented, in such a manner, that, unless to very curious and sagacious persons, the means how such Apparitions are made, shall not be discoverable. The way in short is this;
Opposite to the place or wall, where the Apparition is to be, let a Hole be made of about a foot in diameter, or bigger; if there be a high Window, that hath a Casement in it, 'twill be so much the better. Without this hole, or Casement open'd at a convenient distance, (that it may not be perceived by the Company in the room) place the Picture or Object, which you will represent, inverted, and by means of Looking- glasses placed behind, if the picture be transparent, reflect the rayes of the Sun so, as that they may pass through it towards the place, where it is to be represented; and to the end that no rayes may pass besides it, let the Picture be encompass'd on every side with a board or cloath. If the Object be a Statue, or some living Creature, then it must be very much enlighn'd by casting the Sunbeams on it by Refraction, Reflexion, or both. Between this Object, and the Place where 'tis to be represented, there is to be placed a broad Convex-glass, ground of such a convexity, as that it may represent the Object distinct on the said place; which ​any one, that hath any insight in the Opticks, may easily direct. The nearer it is placed to the object, the more is the Object magnified on the Wall, and the further off; the less; which diversity is effected by Glasses of several spheres. If the Object cannot be inverted (as 'tis pretty difficult to do with Living Animals, Candles, &c.) then there must be two large Glasses of convenient Spheres, and they plac'd at the appropriated distances (which are very easily found by tryals) so as to make the representations erect as well as the Object.
These Objects, Reflecting and Refracting Glasses, and the whole Apparatus; as also the Persons employ'd to order, change and make use of them, must be placed without the said high Window or Hole, so that they may not be perceived by the Spectators in the room; and the whole Operation will be easily perform'd.
The particular manner of preparing the Objects, adapting the Glasses, collecting the Rayes of the Sun, varying the Object, making the representations of the Sky (by the help of other Glasses) and of Clouds (by the help of Smoak) &c. I intend, hereafter, when I have leisure and opportunity, more particularly to describe; as also the way, of making a natural Landskip, &c. to appear upon the walls of a light room; which will not only be very pleasant, but of great use in painting. Whatsoever may be done by means of the Sun-beams in the daytime, the same may be done with much more ease in the night, by the help of torches, lamps, or other bright lights, plac'd about the Objects, according to the several sorts of them.
So far our Inventor; who hath not contented himself with the bare speculation, but put the same in practice some years since, in the presence of several members of the R. Society, among whom the Publisher had the good fortune to see the successful performance of what is here answered.
SIR, You may perchance remember, that some time since, ​there was a discourse at London of a certain pot of Glass-mettal, which brake in the Glass-house at Woolidge; in the bottom of which was found a quantity of Opal-glass. And although the very persons, who had compounded it, endeavoured to repeat that accidental Experiment, yet they could never bring it to pass, as I was inform'd by a person concern'd in it. The last week I was two daies at Harlem on purpose to see the Experiment of the making of this counterfeited Opal-glass which is there done by Rule. It is very lively, I confess, and as I guess, perform'd only by the degrees of heat, producing the Colors; of which degrees I have by me several, I took notice of with some curiosity, in the operation. When the Composition is thoroughly melted, they take out some on the point of an Iron-rod, which being cool'd either in the Air or Water, is colourless and pellucid; but being put into the mouth of the furnace on the same rod, and there turn'd by the hand for a little space, hath its little bodies so variously posited in several parts of the same piece, as that the light falling on them, being variously modified thereby, represents those several Colours, that are seen in the true Opal. Whether it be the greatest, or least degree of heat, that renders it a white opaque Body, l have let slip; but this I know (which seems remarkable) that the colours of it may be destroy'd and restor'd, according to the various motions (I suppose) of its particles by heat.
They also make there the Amethyst and Saphir; and have recover'd the hundred years loss of incorporating Red-glass; and have some mettal, that is esteem'd to equal Chrystal in hardness as well as colour. To give you an account of their Mill to grind, and Engins to polish Looking-glasses, will be needless: I onely add, that they can diamond or square their looking-glasses in their Grinding-mill.
Epistolam tuam, Vir Clarissime, quæ Du Laurensii impressam chartam ​(quæ me spectabat) habuit inclusam, accepi hodie: cui statim respondendum censui; neque enim deliberatione opus est. Expectabas Tu à Me non ita pridem literis tuis, ut, quid Ego de Du Laurensii libro tum nuper edito (quem cum iis literis in eum finem mittebas) sentirem, Tibi paucis exponerem. Quod cum Ego privatis ad Te literis fecerim (quippe hoc an ico expetenti negandum non putavi) Tu harum partem aliquam typis vulgandam paulo post curasti (eam nempe, quæ illatam mihi injurieim expostulabat) reliqua, in sui, credo, gratiam, reticendo. Hæc homini bilem movent.
Quibus Ego haec summatim repono, tum mihi fuisse liberum, amico expetenti, libere, quid sentirem, exponere; tum & Te arbitrio tuo usum esse. Speciatim vero, quod injuriam spectat, quam mihi factam querebar, dum me tanquam Thrasonem aliquem inducit, Problema leviculum (& quidem, prout ipsius verbis, exponitur ridiculum) totius Europae Mathematicis proponentem; quo de me triumphum ageret, monstrando, quam ille potis sit solvere. Non diffitetur errasse se;(& quidem res ipsa clamat; quippe non Ego, sed Montfertius nescio quis Gallus, illud quod innuit Problema Anglis proposuerar, quod varii variis modis solutum dederint; inner quos & Ego.) Hoc tantum causatur, quod Amicus quidam tale quid (non dicit, id ipsum) ipsi retulerat.
Sed quorsum est, ut Amicum advocet, cum, quid Ego ea in re fecerim, jun palam prostet in scriptis meis editis. Et quidem suspicor suum tale quid, quod ab Amico acceperat, non aliud fuisse, quam hujusmodi Problema in scriptis meis à me solutum extare; pro quo ille (pari ac in reliquis negligentia) à me propositum substituit, additque de suo, Totius Europæ Mathematicis, quo & jactantior Thraso, & Triumphus suus sit illustrior, Ut ut sit, hoc eum male habet, quod non simpliciter negaverim, me istius Problematis authorem esse, sed (quod garrulitatem vocat) mea verba cum suis juxta-ponendo, ostenderim, quam mihi manifesto suerit injurius.
De cæteris autem, non placet ei, quod dc suis Ego censuram instituerem, h. e. nollrt, ut Ego tibi petenti dicerem, quid de Libro edito sentirem. Sed quidni liceat? Nam & idem alios sentire, Tu eriam nofii, Male etiameum habet, quod ceusuerim, plus fronte cum polliceri quàm opere absulverit; (nempe hoc mollius sonare putaram, quàm si dixissem, Parturiunt Monte, &c.) Sed & Tu a1ios ]uxta mecum sentire; Fastuoso titulo librum haud satis respondere, non ignoras, nedum in exauctorati Euclidis vices successurum. Neque prius illis fidem faciet, rem secus esse, quàm viderint, Genuina Matheseos Principia & Elementa vera quæ hucusque nondum traditæ insinuat, ab ipso felicius tradi, quam ea tradiderint superiores.
Dixeram, partem magnum ex Oughtredi meisque sriptis (ut ut neutrius meminerit) desumptam VIDERI, (nempe propter multa, quæ nobiscum habet commuma, peculiares loqueudi formulas, ipsaque eadem symbola passim retenta; item ex Victa, Schotenio, aliisque ab eo editis, quorum & subinde meminit. Sed quo animo quove consil1o hæc dixerim, dicit se ​plane non capere. Dicam (imo res ipsa dicit, quippe hoc inde directe sequitur) nempe ut ostenderem Tibi, sua non esse Nova omnia, & hucusque nondum tradita. Rem ipsam quoad cæteros agnoscit (quibus & Hariotum accenset;) Mea tantum scripta dicit se non legisse; quod excusatum petit. (Esto. & habeo excusatum: At interim non eo magis inter hucusque nondum tradita, censenda erunt, quod ipse non legerit.) Et quidem, Oughtredm quod spectac, enumerat aliquam multa, quæ fjm fatetur ex ipso quasi verbatim variis in locis transcripta; atque excusatum it, quod Authorem non nominaverit: (Unde me conjecturam non temere fecisse, satis constat.) Sed negat ea partem magnam ((respectu totius) dicenda esse. (Patior itaque, ut pro parte magna, modo id dictum malit, partem potissimam rescribat.) Verum Ego, non numero verborum, sed return pondere, partem magnam æstimo; nec ea tantum ex Oughtredo desumpta videri existimo, quæ totidem verbis apud cum extant, sed totam eam doctrinam, utut aliis verbis expositam, quæ ab ipso jam ante multos annos tradita fuerit, quamque ex eo hausisse videri possit hic Author; licet hic pluribus forte paginis, quàm ille lineis, rem eandem explicaverit. Id itaque dictum velim, Magnam partem earum RERUM, quæ hic traduntur, apud Oughtredum (ne & mea scripta interponam) vel totidem verbis extare, vel verbis tantundem significantibus, vel inde posse levi negotio deduci; ut non pro Rebus hucusq; nondum traditis censeri debeant. Sed & hinc desumptam (remipsam quod spectat, licet variata nonnunquam verborum formula) videri, propter easdem non raro peculiares loquendi formulas retentas, ipsaque eadem non raro symbola. Quæ quidem aperta sunt vestigia, unde hæc desumpta sint, ut jam non possit ipse non fateri, ut ut nomen prius reticuerit. Atque eadem, servata proportione, de cæteris quos dixeram, inteilige. Non enim Ego utrobivis intenderam crimen Plagii (quod ipse amoliri vellet) sed ut tibi dicerem quod res est, Principia sua, quatenus sana sunt, tum & aliis fuisse pridem cognita, tum & ab aliis dudum tradita, rem ipsam quod spectat, ut ut sub aliis verborum formulis; neque jam primitus detecta, atque hucusque nondum tradita. Sed & Tibi digitum intendi, apud quos Authores hæc eadem negotia reperias ipse, & quidem, prout Ego sentio, non minus feliciter exposita. Qui sententiæ concinit, quem ad me de illo characterem scripto misit Vir quidam Mathematicus, Tibi non ignotus (Du Laurensio, credo, non inferior) priusquam Ego librum videram, nec interrogatus quidem: Algebram (inquit) Du Laurensii, ad D. Oldenburg transmissam, vidi: qui autem Tua, Cartesqiue & hujus interpretum scripta viderunt, Authorem, credo hunc non sunt admiraturi: Quasi quidem Ego non tam Censere dicendus sim, quàm Consentire.
Sed conqueritur porto, quod dixerim, inibi reperiri, aliqua parum sana, & minime accurata multo plura. Quorum alterum jam fatetur ipse (ut no Tibi fuerim hac ex parte iniquus Judex,) alterum non-dum. Neque tam conqueritur quod hæc censuerim, quàm quod hujus censuræ causas in pubiicum non protulerim; quippe hoc fecissem, tum de publico, tum de ​seipso, gratiam (inquit) meruissem. Hoc autem erimine tuum est me levare. Rogatus enim à Te sententiam, Ego datis ad Te literis, & quid censerem, paucis indicavi, & cur ita. Addebam scilicet (non quidem justam totius libri confutationem, neque enim id agebam; sed) pauca specimina eorum, quæ cursim legenti occurrebant vel parum sana, vel minus accurate dicta, Quo autem consilio Tu, cum partem horum in publicum emiseris {quo forte illatim mihi injuriam utcunque elueres) reliquum reticueris (quod in illius gratiam factum putaverim) Tu melius noveris. Quoniam vero & ille hoc expetit, per me licebit, ut tota Epistola, prout scripta fuerat, quæ & tuæ potestatis, (utpote ad Te scripta) facta est, simul prodeat, ut judicet Orbis literatus, num non justas habuerim ita censendi causas, ut ut stricturis brevibus infinuatas; atque resciscac ipse, poiiquam irum decoxerir, quam inibi & libere & candide egerim; lihere tecum, & cum illo, satis candide. Novas ego jam non adjungo, tum quod libcr ipse mihi nunc præ manibus non sit; tum, si esset, non nova hanc ob causam recenfione censerem indigere. Neque enim mihi tunc erat in animo ad vivum omnia resecare, necdum est. Id olim forte fier, fi necelle videbitur; quod non fore autumo quippe non tanti res est. Quod ad Problema spectat, quod non à me Freniclio, ut difficile, propositum innuit, atqie ab ipso solutum; rem secus atque est narrat. Patet utique, Scriptis editis, neque Freniclio `a me propositum fuisse, neque u defficile, Problema quod insinuat, aut etiam ut magni momenti; sed apud alium (cum Ego de Freniclio nihil inaudiveram) obiter infinuatum, tanquam Fermatiano simile; (Vid. Commercium Epistolicum pag. 35. lib. 4. & seqq.) Qxod autem Ego Problems meum depreciatum iveram, arripuit Freniclius, sponte sua, ut fatis elegans, & solutione sua dignum. Quæ quàm aliena sint ab iis, quæ hic narrat Du Laurens; ipse videas, non possum non rogare, ut imposterum velit ille in Historicis enarrandis fidelius agere, atque in tradendis Mathematicis accuratius.
Opprobria, reliquamque quam haber maledicendi copiam non attingo, quoniam hæc non aliud demonstrant magis, quam impotentem scribentis animum, & me minus, quam illum, feriunt. Tu interim, Vir Clarissime, vive & vale.
          Tuus, &c.
Oxonii, 2 Julii, 1668.
Although the Publisher wished very much, that he might not be necessitated to say any more of this subject, after he had (what he thought Justice required of him) made publick Doctor Wallis's Vindication of the Injury done him in the End of Monsieur Du Lauren's Book, here in debate; yet, since this Author in his printed Letter, mentioned in the Title to the precedent Animadversions, presseth hard to know the Reasons of the Doctors ​Censure, which, en passant, he thought fit, being desired, to give of that Book, and maketh the Omission thereof the chief ground of his Complaint in his said Letters, it seems unavoidable to comply with him in that demand, and to publish, what (out of respect to the same) was supprest ever since that Vindication was printed, with which it then came joyned, as follows;
— ——Sed revera (ut quod res est dicam) D. Du Laurens eorum, quæ scribit, negligentior est, quam Mathematicum deceat. Cujus quidem specimim, ne hac vice longius petitum abeamus, in hoc Montfertii problemate, ut à D. Du Laurens exposito, satis suppetunt.
Cur pro extremis Ellipseos diametis (hoc est, maxima & minima, perperam substituat, Diametris maximis (quasi in Elllpsi plures essent maximæ diametri) causa, desidero, quæ oscitantiam excuset.
Similiter; Ubi substituitur in transversa ejus diametro, pro, in Axe transverso (quasi vel Unica esset Diameter transversa vel præter Axes nulla; vel, in quavis indifferenter transversa-diametro assignari punctum, intelligendum esset; etiam cum, præter Axes, nulla sit data.) Dixisset utique pari jure, ubivis intra Ellipsin assignato. quippe nullum est intra Ellipsin punctum, quod not sit in aliqua transversa-diametro.
Insuper; cum imperatum sit, ut, quæ Requiruntur, Numeris exhiheantur, consentaneum esset, ut & quæ Dari perhibentur etiam Numeris Data essent. Adeoque pro, Datis Ellypseos Diametris maximis, dixisset potius, Ellipseos Diametrius Extrermis (non maximis,) per numeros designatis vel in numeris datis. Item, pro, tum assignato puncto in transversa ejus Diametro (ubi, puncto in numeris data, minus conveniret;) potius dixisset, punctoque in utrovis Axe transverso (non transversa diametro) per suam vel à Cenrto, vel à Vertice, distancium, numero designatam, assignato. Item, pro, Segmenta lineæ intra Ellypsin terminatæ (quod neutrum vel Lineæ, vel Segmentorum ejus, extremum determinat;) dixisset potius, Segmenta rectæ, Ellipsi (non, intra Ellypsin) terminatæ, in puncto illo fectæ; vel, Segmenta rectæ per punctum illud transeuntis, huic Axi (seu Puncto) & Ellipsi interjecta; vel, rectae Segmenta. Ellipsi & Puncto illo terminata; vel, quod sit ίσοδόυαμγ, quod tum rectæ extrema, yum punctum Sectionis designaret; quorum neutra ipsius verbis determinantur, sed conjecturæ permittuntur.
Atque hæc in una Propositione (eaque non longq) tam multiplex incuria, eo minus veniam meretur, quoniam Montfertii Problema, quod Du Laurens tam imperfecte recitat, multo felicius exaratum erat, quud itaq; D. Du Laurens vel in melius mutasset, vel non mutasset. Undecunque enim hoc Problema desumpserit (sive ex typis edito Montfertii proplemate, sive ex Wrennii solutione, typis item edita, sive ex meis editis libris) non potuit non videre Problema illud felicius conceptum; sed &, quod Jean de Montfert (non Johannis Wallisius) proposuerat.
​Sed &, quæ sequuntur, simili laborant negligentia. Cum enim preposite quæstionis solutionem aque facilem in numeris, ac in liesis, prædicat, (adeoque peritiam' suam eo commendatam innuit, quod & in Lineis præitare id possit) id omnino secus est. Quó enim id in lineis fiat, nihil difficilius requiritur, quam ut quis rectam ducat quæ expositam in datis angulis secet; (quippe si recta sic dicatur, quæ in assignato puncto Axem ita secet, Ellipsis ipsa, absque nova constructione, segmenta determinabit.)
Etiam hoc addas licet, quod, cum propositum sit segmenta reperire, processus ejus non-nisi Segmentum majus exhibet; non autem, vel minus segmentam, vel tot am lineam. Verum quidem est, Segmentum minus facile repertum iti (sed & majus non minus facile;) at quomodo id sieret, cum hoc susceperit, innuisse oportebat.
Problematis hujus ad alias sectiones Conicas accommodacio, non est nova difficultas: Idem enim processus, quem ad Ellipsin indico; etiam ad alias curvas, quarum Ordinatim-applicatæ pariter innotescant, mutatis mutandis, accommodabitur; (id sive intra curvam, sive extra, Diametrum secet sic ducta recta). Atque harum rerum vel modice peritus, prout casus tulerit, rem accommodabit.
Nolo Tibi molestus esse tam fedula negligentiarum, quæ per totum librum occurrunt, enumeratione: Non utique Confutationem scribo sed specimina negligentiæ; tantufque illarum in tam non longo Problemate numerus susficiat.
Adjungam tamen pauca earum, quæ in opera occurrunt, negligentiarum specimina. Pag. 67. Ubi, Ut duas Rectas spatium non comprehendere, sic, neque duo Plana, pronunciat. Recte quidem: sed neque Plana tria; hoc utique dicendum erat. Ut enim Rectæ tribus pauciores suerliciale spatium, ita nec Plana pauciora, quam quatuor, solidum concludent. Sed neque admodum accuratum est, quod proximis verbis subjangit, Plana due no duobus locis fibi occurrere; quippe in omnibus locis illius Rectæ, quæ est communis eorum Sectio, occurrunt invicem; non autem extra illam.
Verum (in eadem pag) quæ sequitur Anguli definitio, negligentior adhuc est, & parum sana: Qui definitur, Duarim pluriumve, ejusdem speciei, magnirudinum (ad unum punctum collectarum, &c.) brevissima distantia. Nam (ut de Tempore, Pondere, Viribus, &c. taceam, quæ tamen ipse alibi pro magnitudinibus agnoscit, funtque, sensu Euclideo, υξγίσπ reputanda;) sumpya sensu stricto magnitudinis voce, pro extensa magnitudine corporea, agnoscit ipse, solida excipienda: adeoque pag. 89. hujus habet Retractationem.
Sed &, pro ejusdem speciei, dixisset potius, ejusdem Generis (quippe hoc est quod vellet) hoc est, Magnitudinem Homogemearum. Nam linea Recta & Curva (cum homogeneæ sint, feu ejusdem generis,) ut ut non sint ejusdem speciei, Angulum constituunt: sic & superficies specie differentes. Verum & Recta cum Plano (aliave superficie) suum habet lnclinationis angulum (non minus quam duo plana) ut ut sint Heterogenea.
​Porro, cum Angulum sic, ut dictum est, definiverat, p. 67; subjungit, p. 68. Quodsi magnit udines illæ sint duæ lineæ, comprehensus ab iis angulus, Planus vocabitur: quasi quidem de Triangulis sphæricis nil unquam inaudivertt; nec alius esse possit superficialis angulus, quam in Plano,
Adhæc, illud duarum pluriumve, de Lineis non tuto dicitur. Trium enim linearum concursus, non angulum, sed angulos saltem duos, constituunt; non enim lineæ plures duabus ad unum superficialem angulum constituendum concurrunt. Item, cum p. 67. Angulum in genere per duarum pluriumve, &c. definiverat; Angulum p. 68. una vel pluribus superficiebus comprehensum ait (& unia quidem angulum verticalem Coni comprehendsum;) quasi quidem una, fuerit, duæ vel plures.
Insuper, quid demum illud est, quod per brevissimam distantiam insinuatum vult? Quippe in ipso concursus puncto, Nulla est distantia; extra illud, nulla minima: nulla utique assignari poterit, qua non sit minor: sed revera tota hæc, quam de Angulo notionem concipit, est param sana. Definiendus utique est non per distantiam seu remotiionem, sed per Inclinationem. quod ex Euclidis definitione didicisset.
Deniq; (ne multus nunc sim) p. 171. in duabus his Quadraticarum æquationum formulis aa−ca+dd==o, & aa + ca + dd ==o; utramque radicem affirmativam esse pronunciat. quod omnino secus est. Et quidem in priore, Radix utraque Affirmativa; sed in posteriore, Negativa utraque.
Atque hæc quadem, ex multis pauca, si non sufficiant, ut ex ungue Leonem æstimes, plura facile congerentur. Num autem hos Incuria, an Inscitia, errores fuderit (orout ipse pag. ult. distinguit) non determine.Vale.
Hæc Dn. Wallisius epiliola una; cui postea submisit alteram, 18. Julii ad me scriptam, quam istoc mense, ob alia, non licebat typis committere; nec quidem licet hoc ipso: ne scil. hasce schedulas, publicationi variorum, idque imprimis sermone Anglico, destinatas, disceptationibus Latinis compleamus. Proxima occasione, quæ idem Author porro notanda invenit vel in unico primo Capite Synopseos Laurentianæ, Lectori (cum particularia flagitet Dn. Du Laurens) ob oculos sistemus.
An Account of Two Books.
This Treatise was promised by the Author in a printed Epistle of his, which we gave an account of in April last, Num. 34. p. 663. There being at the same time publisht a Prodromus of Job. Van Horne, suspecting, that the Observations of De Graef were much the same with his upon this Subject; we do now upon the perusal of this Book, find chiefly these considerable Differences between them.
​First, the said Van Horne makes the Spermatick Artery in man to goe to the Testiclcs in a winding, but De Gaaf, in a streight way.
Secondly, the former affirms, that the vasa deferentia have no communication with the vesiculæ seminales; but the latter maintains, and demonstrateth it to the Ey, there is so great a commerce betwixt them, ut semen dum à Testibus per vasa differentia affluens in Urethram efftuere nequit, propter carunculam clausam; necessariò in influat in Vesiculas, in iisque pro futuro coitureservetur.
Thirdly, the former is of opinion, triplicem esse materiam seminis; but De Graaf will have but one only; answering the Arguments, used both by Van Horne and Dr. Wharton to prove that triplicity.
But that, which De Graaf much insists on in this Book, is, to shew what is the true Substance of the Testicles, and to vindicate the Discovery thereof to himself; affirming positively, that no man, before him ever knew the truth of it.See the Letter of Doctor Tim. Clark, N. 35. p. 681 For the making out of which, he first denyeth, that the Testes are glandulous, or pultaceous; and then affirms that their substance is nothing else but a Congeries minutissimorum vasculorum semen conficientiam, quæ si absque ruptione dissoluta sibi invicem adneiteretur, facile viginti ulnarum longitudinem excederent. Which he affirms, he can grove by ocular Demonstration.
Then he sheweth, how the seminal vessels pass è Testibus ad Epididymides, vid, not by one Trunck (as Dr. Highmore thinks) but by 6. or 7. small ductus's; assigning the cause, why Doctor Highmore did not see them.
Further he examines, An semen in testibus conficiatur; utrum ex Sanguine vel ex Lympha; quomodo elaboretur, craisescat, lactescat: qua via à Testibus ad Urethram excurrat.
Moreover he endeavours to prove, Vesiculas seminales ordinatas esse non seminis generationi, sed receptieni & asservationi.
He also observeth concerning the seminal matter, that 'tis composed ex duplici materia, which after Aristotle, he calls, λόγος ωερμαπχόυ χαί mi Byron an-eggaaflruév, considering this twofold matter like Dough and Ferment, this infecting and quickening that, and the grosser part being a conservatory and vehicle to that, which is most elaborate.
When he examine the Penis, he taketh notice, Omnes ​hastenus Anatomicos perperam assignasse usum musculorum Penis, quos Erectores apprellant; Eorum quippe provinciam non esse, Penem erigere, & dilatare Urethram, cum omnis Musculi actio sit rontracto, quæ extensioni contraria est; eos potius Penem versus interiora retrahere quam eriqere: Interim, hosce Penis Musculos, coarctando corpora nervosa circa corum exortum, materiam seminalem versus Penis partem anteriorem propellere, atque hac ratione corporum nervosorum distensione erectionem augere.
Before we conclude this Account, we cannot but take notice, that the Author occasionally inserts in this Book divers curious and remarkable Examples and Observations; some whereof are.
1. Concerning those, that are born, either absque Testibus; or, cum Testiculo uno; or, cum tribus, idque hæreditario per aliquot familias, admodum fæcundas.
2. About the situs præternaturalis Testiculorum, generationis tamen virtutem non impedientis.
3. Concerning lactescent Bloud in a man living at Delft in Holland, whose Bloud alwayes turn'd into Milk, when let out either by venæ-sections, or by bleeding at the Nose, or by a wound V. pag. 84, 85. Compare Numb. 6. pag. 100, 117, 118. and Numb. 8. pag. 139. of these Transactions.
4. Concerning the strange alteration made in Femals, ab Aura seminali: quod confirmat exemplo felis, diu sugentis (idque ad integram fere sui nutritionem) lac mammarum caniculæ, per aliquot annos à coitu prohibitæ; deinceps vero, postquam catella admiserat canem, nunquam ab eo tempore lac ex mammis ejus exsugere volentis,
5. About a strange Hæmorehagy per Penem, which amounted to 14. pound, in a Porter of 52, years old, falling down with a heavy load upon a board, laid over a ditch, and so turning about, when the said porter trod upon it, that it cast him down upon its edge, turn'd between his legs; yet the Patient by the skill and care of our Author recover'd.
6. Various Observations of Clysters and Suppositories, cast up by Vomits, p. 195, 196.
7. Several wayes of performing unbloudy dissections of Animals, p. 228, 229, &c.
Incidebam heri (illustrissime Domine) in D. Mercatoris Logarithmotechniam, nuper editam. Quæ ita mihi placuit, ut non prius dimiserim quàm perlegissem totam. Et quamquam pauca quædam, Phraseologiam quod spectat seu loquendi formulas nonnullas, mutata mallem; sunt tamen ipsa sensu suo sana; Eaque quæ superstruitur Doctrina, Logarithmos expedite atque subtiliter construendi, perspicue satis atque ingeniose traditur.
Quae huic subjungitur, Quadraturæ Hyperbolæ, elegans admodum est atque ingeniosa. Nempe ad hunc sensum. V. Fig. 1.
$MBF$
$AN$
$AH$
$BIA$
$FHA$
$spA$
$BI$
$FH$
$sp$
$AN$
$AI=BI=1$
$HI=a$
$FH={\frac {1}{1+a}}$
$AH.AI::BI.FH.$
$1+a.1::1.{\frac {1}{1+a.}}$
$1$
$1+a$
${\frac {1}{1+a}}=1-a+a^{2}-a^{3}+a^{4}$
$1+a)1(1,-aa^{2},+,a^{3},+,a^{4},\mathrm {\&c}$ :
| $+a$ | | | | |
| $-a$ | | | | |
| $-a$ | $-a^{2}$ | | | |
| | $+a^{2}$ | | | |
| | $+a^{2}$ | $+a^{3}$ | | |
| | | $-a^{3}$ | | |
| | | $-a^{3}$ | $a^{4}$ | |
| | | | $+$ | $a^{4}$ |
| | | | | $\mathrm {\&c.}$ |
Cumque hoc perinde obtineat, ubicunque ultra punctum I, ponatur $H$. Positis, ut prius $AI=a$; hujusque continuatione qualibet, ut $Ir=A$; quæ intelligatur in æquales partes innumeras dividi, quarum quælibet, ut Ip. pq, &c. dicatur a; adeoque Ip, Iq, &c, sint a, 2a, 3a, &c. usque ad A: Quæ his respondent rectæ ps, qt, &c. usque ad ru, (spatium BI ru complentes) sunt.
| $1-a+a^{2}-a^{3}$ | $+a^{4}\;\mathrm {\&c.}$ |
| $1-2a+4a^{2}-8a^{3}$ | $+16a^{4}\;\mathrm {\&c.}$ |
| $1-3a+9^{2}-27a^{3}$ | $+81a^{4}\;\mathrm {\&c.}$ |
| & sic deinceps usque ad ||
| $1-A+A^{2}-A^{3}$ | $+A^{4}\,\mathrm {\&c.}$ |
| Cum itaque sint | $1+$ $1$ $+1\mathrm {\&c.}$ (usque ad ultimum) $=A$ $a+$ $2a$ $+3a\mathrm {\&c.}$ (usque ad $A$) $={\frac {1}{2}}A^{2}$ $a^{2}+$ $4a^{2}$ $+9a^{2}\mathrm {\&c.}$ (usque ad $A^{2}$) $={\frac {1}{3}}A^{3}$ $a^{3}+$ $8a^{3}$ $+2a^{3}\mathrm {,\;\&c.}$ (usque ad $A^{3}$) $={\frac {1}{4}}A^{4}$ & sic deinceps: |
(quod ostendit ille prop. 16, estque à me alibi demonstratum:) Recte colligit, prop. 17. Expositum spatium Hyperbolicum $BIru=A-{\frac {1}{2}}A^{2}+{\frac {1}{3}}A^{3}-{\frac {1}{4}}A^{4}+{\frac {1}{5}}A^{5}$, &c. Adeoque si (assignato, ipsi $A=Ir$, valore suo in numeris, ut res postulaverit,) distribuantur ih duas classes $A$, ${\frac {1}{3}}A^{3}$, ${\frac {1}{5}}A^{5}$, &c. (potestates affirmatæ,) & ${\frac {1}{2}}A^{2}$, ${\frac {1}{4}}A^{4}$, &c. (potestates negatæ;) harumque Aggregatum, ex Aggregato illarum, subducatur; Residuum erit ipsum $BIru$ spatium Hyperbolicum.
Nequis autem operam lusum iri existimet,, propter Addendorum seriem in utraque classe infinitam; adeoque non absolvendam: Hinc incommodo medelam (tacitus) adhibet: ponendo $A=0.\;1$, vel $A=0$, 21, aliive fractioni decimali æqualem, adeoque minorem quam $1$: (Hoc est, sumpta $Ir$ minore quam $AI=1$.) Quo fit, ut postriores ipsius $A$ potesiates tot gradibus infra Integrorum sedem descendant, ut merito negligi possint.
Exempli gratia; positis $AI=1$, & $IR=0$ 2i. erit
| | $A$ | $=0,21$ |
| ${\frac {1}{3}}$ | $A^{3}$ | $=0,003087$ | ${\frac {1}{2}}$ | $A^{2}$ | $=0,02205$ |
| ${\frac {1}{5}}$ | $A^{5}$ | $=0,0000081682$ | ${\frac {1}{4}}$ | $A^{4}$ | $=0,000486202$ |
| ${\frac {1}{7}}$ | $A^{7}$ | $=0,000002572$ | ${\frac {1}{6}}$ | $A^{6}$ | $=0,000014294$ |
| ${\frac {1}{9}}$ | $A^{9}$ | $=0,000000088$ | ${\frac {1}{8}}$ | $A^{8}$ | $=0,000000472$ |
| ${\frac {1}{11}}$ | $A^{11}$ | $=0,000000003$ | ${\frac {1}{10}}$ | $A^{10}$ | $=0,000000016$ |
| | $+$ | $0,213171345$——— | | | $0,022550984$ | $=0,190620361=BIru$ |
Quæ est brevis Synopsis Quadraturæ suæ satis elegans.
Dissimulandum interim non est; sequis totius $BIHF$ spatii (cujus latus $IH$ longius intelligatur quam $AI$) quadraturam postulet; rem non ita feliciter successuram: propter medelam, quam modo diximus, malo minus sufficientem. Cum enim jam ponenda $A>1$; manifestum est, posteriores ipsius potestates, altius in Integrorum sedes penetraturas, adeoque minime negligendas.
Huic autem incommodo, levi constructionis immutatione, facile subvenitur. Vid. Fig. 1.
Cæteris utique ut prius constructis; Quadrandum exponatur $HF$ ur ​spatium; (cujuscunque fuerit longitudinis $AH$; puta major minorve quam $AI$, vel huic æqualis: sumptoque ubivis inter $A$ & $H$, puncto $r$; puta ultra citrave punctum $I$, vel in ipso $I$ puncto:) Ponantur autem (non, ut prius $AI=1$, & $Ir=A$ : sed) $AH=1$; & $Hr=A$ quæ intelligatur in æquales partes innumeras dividi, quarum quælibet sit $a$. Erunt iraque, post $AH=1$, reliquæ deinceps decrescentes $1-a$, $1-2a$, $1-3a$, &c. usque ad $Ar=1-A$. Item, propter æqualia Rectangula $FHA$, $urA$, $BIA$, &c. puta, $=b^{2}$: Erit $HF={\frac {b^{2}}{1}}$ reliquæque deinceps ${\frac {b^{2}}{1-a}}$, ${\frac {b^{2}}{1-2a}}$, ${\frac {b^{2}}{1-3a}}$, &c, usque ad $ru={\frac {b^{2}}{1-A}}$ spatium $HFur$ complentes. (Quæ omnia ostensa sunt, in mea Arithmetica Infinitorum, prop. 88, 94, 95.)
$1-a)b^{2}(b^{2},+b^{2}a,+b^{2}a,+b^{2}a,\mathrm {\&c}$ :
| $-b^{2}a$ | | | | |
| $+b^{2}a$ | | | | |
| $+b^{2}a$ | $-b^{2}a^{2}$ | | | |
| | $+b^{2}a^{2}$ | | | |
| | $+b^{2}a^{2}$ | $-b^{2}a^{3}$ | | |
| | | $+b^{2}a^{3}$ | | |
| | | $+b^{2}a^{3}$ | $-b^{2}a^{4}$ | $-b^{2}a^{4}$ |
| | | | $+$ | $b^{2}a^{4}$ |
| | | | | $\mathrm {\&c.}$ |
${\frac {b^{2}}{1-a}}=b^{2}+b^{2}a+b^{2}a^{2}+b^{2}a^{3}+b^{2}a^{4}$
Hoc est, $b^{2}$ in $1+a+a^{3}+a^{2}+a^{4}$, &c. (sumptis ipsius $a$ potestatibus continue sequentibus affirmatis omnibus.) Cumque de reiquis idem sit judicium; erunt rectæ omnes, ipsis $HF$ & $ru$ interjectæ,
| | $1+a$ | $+a^{2}$ | $+a^{3}$ | $+a^{4}$ &c. | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$ in $b^{2}$, |
| | $1+2a$ | $+4a^{2}$ | $+8a^{3}$ | $+16a^{4}$ &c. | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$ in $b^{2}$, |
| | $1+3a$ | $+9a^{2}$ | $+27a^{3}$ | $+81a^{4}$ &c. | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$ in $b^{2}$, |
| | & sic deinceps usque ad | & sic deinceps usque ad | & sic deinceps usque ad | & sic deinceps usque ad | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$ in $b^{2}$, |
| | $1+A$ | $+A^{2}$ | $+A^{3}$ | $+A^{4}$ &c. | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$ in $b^{2}$, |
| Omniumq; Aggregatu, | $A+{\frac {1}{2}}A^{2}$ | $+{\frac {1}{3}}A^{3}$ | $+{\frac {1}{4}}A^{4}$ | $+{\frac {1}{5}}A^{5}$ &c, in $b^{2}=FHru$. | |
| | | | | (per Arithm. Infin. prop. 65.) | |
Exempli gratia. :
| Positis $AH$ | $=1$. |
| $Hr=A$ | $=0,21$ |
| $AI=b$ | $=0,\;1$ |
| Adeoque $b^{2}$ | $=0,01$ |
| Erunt | $A$ | $=0,21$ |
| | ${\frac {1}{2}}A^{2}$ | $=0,02205$ |
| | ${\frac {1}{3}}A^{3}$ | $=0,003087$ |
| | ${\frac {1}{4}}A^{4}$ | $=0,00048623-$ |
| | ${\frac {1}{5}}A^{5}$ | $=0,000081682+$ |
| | ${\frac {1}{6}}A^{6}$ | $=0,000014294+$ |
| | ${\frac {1}{7}}A^{7}$ | $=0,000002573-$ |
| | ${\frac {1}{8}}A^{8}$ | $=0,000000473-$ |
| | ${\frac {1}{9}}A^{9}$ | $=0,000000088+$ |
| | ${\frac {1}{10}}A^{10}$ | $=0,0000000017-$ |
| | ${\frac {1}{11}}A^{11}$ | $=0,000000003+$ | |
| Horum summa | Horum summa | $-0,235722333$ |
| Ducta in $b^{2}$ | Ducta in $b^{2}$ | $=0,01$ |
| Exhibet —— | Exhibet —— | $-0,0023572233=FHRU$ | $-0,0023572233=FHRU$ |
| Qualium $1.=ANGN$ $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ | Quadrato, | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ si angulus $A$ sit $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ | Rectus. |
| Qualium $1.=ANGN$ $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ | Rhombo, | $\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,$ si angulus $A$ sit $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ | Obliquus. |
Quæ quidem tam absoluta est tamque expedita Hyperbolæ quadratura, ut nesciam an melior sperari debeat.
Atque hæc sunt quæ hac de re scribenda duxi. Quæ si D. Mercatori impertiveris; non displicebit, credo, hæc suæ Quadraturæ facta accessio.
Posse hæc ad Logarithmorum inventionem accommodari, non est quod moueam: Sed & ad summam Logarithmorum inveniendum: (quam inquirit ille prop. 19.) Nempe, Positis $AH=1$, $AI=IB=b$, (ut prius) planoque $BIHF=pl.$ Erit $pl-b^{2}+b^{3}=BIps+BIqt+BIru$, &c. usque ad $BIHF$. Si autem non ab ipsa $BI$ incipiatur; sed ultra citrave, puta $2ps$: Posita $pH=a$ & $psFH=pl$. erit (universaliter) $pstq+psur$&c. (usque ad $psFH$) $=pl-ab^{2}$: qualium 1, æquetur cubo ipsius $AH$.) Quod alias, si opus exit, demonstrabitur. Tu interim, illustrissime Domine, Vale.
    Oxon. d. 8. Julii, 1668.
| Fig. 1 | Fig 2 |
The Demonstration
Promised at the end of the foregoing Letter, follows in another from the same Author to the same Noble Lord, thus;
Petis (Illustrissime Domine) per literas tuas Aug. 3. datas (quas hesterna nocte accepi) ut demonstrare velim methodum meam, Logarthimorum summam inveniendi, quam literis meis Julii 8. datis, brevissime insinuaveram.
Quæ quidem, cum sit cum Ungularum doctrina (quam alibi trado) connexa opus erit ut utramque simul exponam: sed & rem totam (quam D. ​Mercatoris figuræ & methodo quantum res ferebat accomodaveram) ad principia mea revocatam ab origins repetam. V. Fig. 2.
Ostensum est, in mea Arithmetica Infinitorum, prop. 95.. Spatium Hyperbolium $AD\beta \beta \gamma$ (in infinitum continuarum à parte $\beta \delta$, sed à parte $D\beta$ ubivis terminatum,) Figuram esse quam ex Primanorum Reciprocis conflatam appello, Prop. 88. definitam: Cujus nempe Ordinatim—applicatæ $d\beta$, $d\beta$, sint Primanis (seu Arithmetice proportionaiibus) $db$, $db$, (Triangulum complentibus) adeoque ipsis $dA$, $dA$, (suis à vertice distantiis) Reciproce Proportionales. Hoc est, (posito $AD=d$; & rectangulo $AD\beta =b^{2}$; particulisque infinite exiguis $a$, $a$, &c;) fi à vertice $A\delta$ incipias ${\frac {b^{2}}{0}}$, ${\frac {b^{2}}{a}}$, ${\frac {b^{2}}{2a}}$, ${\frac {b^{2}}{3a}}$ &c. usque ad ${\frac {b^{2}}{d}}=D\delta$: vel, fi à base $D\delta$ incipias, ${\frac {b^{2}}{d}}$, ${\frac {b^{2}}{d-a}}$, ${\frac {b^{2}}{d-2a}}$, ${\frac {b^{2}}{d-3a}}$, &c. usque ad ${\frac {b^{2}}{d-d}}=A\delta$ infinitæ, (nempe, fi ad Verticem usque processum continuaveris;) vel, usque ad ${\frac {b^{2}}{d-A}}=C\beta$, (posito $DC=A$,) fi continuaveriis usque ad $C\beta$, ubivis intra $A\delta$ & $D\beta$ sumptam. (Adeoque omnium Aggregatum; ${\frac {b^{2}}{d}}+{\frac {b^{2}}{d-a}}+{\frac {b^{2}}{d-2a}}+{\frac {b^{2}}{d-3a}}$, &c, est ipsum $DC\beta \beta$ planum.)
Manifestum itaque est, (& ibidem pro. 94. ostensum) si intelligantur singulæ $d\beta$, in fuas à vertice distantias $Ad$, ductæ; hoc est, ${\frac {B^{2}}{a}}$ in $a$, ${\frac {B^{2}}{2a}}$in $2a$, (& sic de reliquis;) crunt omnia rectangula $Ad\beta$; hoc est, rectarum $d\beta$ momenta respectu $A\delta$, (intellige, facta ex magnitudine in distantiam ductæ;) seu plana semiquadrantalem Ungulam (cujus acies $A\delta$ complentia, (eisdem $d\beta$ rectis perpendiculariter insistentia;) invicem æqualia. Quippe singula $=b^{2}$. (Quorum cum unum sit $AIV\delta$ quadratum, erit $AI=b$.)
Adeoque Totius $AD\beta \beta \delta$ (plani infiniti) seu omnium $d\beta$ illud complentium, momentum respectu rectæ $A\delta$, (ut axis æquilibrii;) seu Ungula semiquadrntalis eidem $AD\beta \beta \delta$ insistens (aciem habens $A\delta$;) sunt totidem $b^{2}$; hoc est, $db^{2}$. (Ungula magnitudinis finitæ plano infinitæ magnitudinis insistens.) Ejusque pars plano $AC\beta \beta \delta$ insistens (propter $AC=d-A$.) similiter ostendetur æqualis ipsi $d-A$ in $b^{2}$. ductæ; hoc est $db^{2}-Ab^{2}$. Adeoque pars reliqua, ipsi $DC\beta \beta$ insistens, æqualis ipsi $Ab^{2}$. Quod itaque est ejusdem $DC\beta \beta$ momentum respectu $A\delta$.
​Atque hoc momentum per plani $DC\beta \beta$ magnitudinem, puta per $pl$, divisum; exhibet plani distantiam Centri gravitatis ab $A\delta$, ${\frac {ab^{2}}{pl}}$: adeoque distantiam ejusdem $aD\beta$, $d-{\frac {ab^{2}}{pl}}$.
Hæc itaque à $D\beta$ distantia, in $pl$ (plani magnitudinem) ducta; exhibet $dpl-Ab^{2}$ ejusdem $DC\beta \beta$ is momentum respectu $D\beta$; feu Ungulam eidem $DC\beta \beta$ insistentem, cujus acies sit $D\beta$.
Hæc denique Ungula (cujus altitude, in $D\beta$, nulla sit, sed, in $C\beta$, ipsi $DC$ æqualis:) fi ex planis ipsi $DC\beta \beta$ parallelis conflari intelligitur; e unt ea, $CD\beta \beta$, $Cd\beta \beta$, & sic deinceps; hoc est, aggregatum omnium $Cd\beta \beta$, $Cd\beta \beta$, usque ad $CD\beta \beta$.
Sunt autem ea plum (ut ex Gregaoii de Sanctio Vincentio, aliorumque post illum, doctrina constat) tanquam Logarithmi Arithmetice proportionalium $Cd$, $Cd$, &c. usque ad $CD$; (feu $a$, $2a$, $3a$, &c. usque ad). Ergo Ungula ipsa, est eorundem Aggregatum. Hoc est (posito $D=1$,) $dpl-Ab^{2}=pl-Ab^{2}$. Quod ostendendum erat.
Porro; cum sit ${\frac {b^{2}}{d-a}}(=d\beta )={\frac {b^{2}}{d}}+{\frac {ab^{2}}{d^{2}}}+{\frac {a^{2}b^{2}}{d^{3}}}+{\frac {a^{3}b^{2}}{d^{4}}}$ &c. (Quod dividendo $b^{2}$ per $d-a$, patebit;) vel, posito $d=1$, (quó ipsius $d$ potestates omnes deleantur,) $b^{2}+ab^{2}+a^{2}b^{2}+a^{3}b^{2}$ &c. seu $1+a+a^{2}+a^{3}$, &. in $b^{2}$. & similiter ${\frac {b^{2}}{d-2a}}={\frac {b^{2}}{d}}+{\frac {2ab^{2}}{d^{2}}}+{\frac {4a^{2}b^{2}}{d^{3}}}+{\frac {8a^{3}b^{2}}{d^{4}}}\;\mathrm {\&c.} =b^{2}+2ab^{2}+4a^{2}b^{2}+8a^{2}B^{2}\;\mathrm {\&c.} =b^{2}$ in $1+2a+4a^{2}+8a^{3},\;\mathrm {\&c.}$ & similiter in reliquis:
| Erunt omnes $d\beta$, (spatium $DC\beta \beta$ complectentes,) | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}$ | $1+a$ | $+a^{2}$ | $+a^{3}$ | $+a^{4}$ &c. | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| Erunt omnes $d\beta$, (spatium $DC\beta \beta$ complectentes,) | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}$ | $1+2a$ | $+4a^{2}$ | $+8a^{3}$ | $+16a^{4}$ &c. | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| Erunt omnes $d\beta$, (spatium $DC\beta \beta$ complectentes,) | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}$ | $1+3a$ | $+9a^{2}$ | $+27a^{3}$ | $+81a^{4}$ &c. | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$ | in $b^{2}$. |
| Adeoq; (per Arithm. Infin. prop. 64.) omnium aggregatum, seu ipsum $DC\beta \beta$ spatium, erit —————— | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}$ | & sic deinceps usque ad | & sic deinceps usque ad | & sic deinceps usque ad | & sic deinceps usque ad | | | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| Adeoq; (per Arithm. Infin. prop. 64.) omnium aggregatum, seu ipsum $DC\beta \beta$ spatium, erit —————— | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}$ | $1+A$ | $+A^{2}$ | $+A^{3}$ | $+A^{4}$ &c. | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| Adeoq; (per Arithm. Infin. prop. 64.) omnium aggregatum, seu ipsum $DC\beta \beta$ spatium, erit —————— | | $A+{\frac {1}{2}}A^{2}$ | $+{\frac {1}{3}}A^{3}$ | $+{\frac {1}{4}}A^{4}$ | $+{\frac {1}{5}}A^{5}$ &c, in $b^{2}=pl$. | $+{\frac {1}{5}}A^{5}$ &c, in $b^{2}=pl$. | $+{\frac {1}{5}}A^{5}$ &c, in $b^{2}=pl$. |
| | | Qualium $1=ABE\delta$ Quadrato vel Rhombo | Qualium $1=ABE\delta$ Quadrato vel Rhombo | Qualium $1=ABE\delta$ Quadrato vel Rhombo | Qualium $1=ABE\delta$ Quadrato vel Rhombo | Qualium $1=ABE\delta$ Quadrato vel Rhombo |
Ideoque, Plani $DC\beta \beta$ momentum respectu $D\beta$; seu semiquadrantalis Ungula eidem insistens cujus acies sit $D\beta$; seu planorum aggregatum ipsam constituentium; seu Logarithmorum summa quos ea representant, $dpl-Ab^{2}=pl-Ab^{2}$, $={\frac {1}{2}}A^{2}+{\frac {1}{3}}A^{3}+{\frac {1}{4}}A^{4}+{\frac {1}{5}}A^{5}$ in $b^{2}$:
​Qualium Cubus (seu Rhombus solidus) $ADE\delta$ sit 1. Si veto non ponatur $d=1$, sed cujuscunque magnitudinis: erit saltem ${\frac {A}{d}}+{\frac {A^{2}}{2d^{2}}}+{\frac {A^{3}}{3d^{2}}}+{\frac {A^{4}}{4d^{4}}}$ &c. in $b^{2}.=pl$.
Vel (posito ${\frac {A}{d}}=e$) erit $e+{\frac {1}{2}}e^{2}+{\frac {1}{3}}e^{3}+{\frac {1}{4}}e^{4}$ &c. in $b^{2}=pl$. Qualium $d^{2}=ADE\delta$ Quadrato vel Rhombo.
Ungulaque (ut prius) $dpl-Ab^{2}$. Qualium $d^{3}=ADE\delta$ Cubo, vel (si angulus $A$ sit obliquus) Rhombo solido.
Cumque $A$. (posito $d=1$) vel $e$ (quicunque ponatur valor ipsius $d$) si minor quam $1$, (propter $A<d$:) illius potestates posteriores ita continue decrescunt, ut tandem negligi possint; planique valor $pl$. exhibeatur quantumlibet vero propinquus.
Atque hæc est, Illustrissime Domine, Methodi, quam innuebnm, ex meis principiis deductio, & demonstratio brevis. Vale. Oxon. d. 5. Aug. 1668.
Some Illustration
Si quorum in manus incidit Logarithmotechnia mea, non inviti, opinor, adspicient paucula hæc exempla, miram istius methodi facilitatem cum summa præcisione conjunctam ostendentia.
nizacls ordo I nCnf¢$ ' u | Expo-'l I - ~l | - W . Bmarii ordo I f ll if 2 5z § Lr'o.5' 4 1 in QW; 1 4.333353 8 -Qléoaeéof y 'L °'f S oo lflfé .fl -, 5;If€,2 32 é,4 I
    6 51; ¢>, |66é66 64 16, 666666;
7, l 1; I4;8§ 7£ 128 lé,28§ V;]4€ I 8 ll If o, iz;; 2.56 32 o g *A , Q 9;I!é|IIlIIl 512 5»», sssss8 5* xo !;} fi, I 1oz4. l l01,4 I P'x>P1> =
Ex primo ordine
| 1 |
| | 0 | 5 |
| | | 0 | 33333333 | 33333333 |
| | | | 02 | 5 | 5 | 5 |
| | | | | 0 | 2 |
| | | | | | 0 | 16666 | 16666 |
| | | | | | | 0 | 1428 | 1428 |
| | | | | | | | 0 | 125 |
| | | | | | | | | 0 | 11 |
| | | | | | | | | | 01 |
| + | 10033534772 | 10033534772 | 10033534772 | 10033534772 | 10033534772 |
| - | 502516792 | 502516792 | 502516792 | 502516792 | 502516792 |
| | 10536051564 | 10536051564 | 10536051564 | 10536051564 | 10536051564 | ⁠9/10⁠ |
| | 9531017980 | 9531017980 | 9531017980 | 9531017980 | 9531017980 | ⁠10/11⁠ |
| 1 |
| . | 05 |
Ex secundo ordine.
| 1 | |
| 2 | |
| 3 | 1 + 2 |
1 n as .oS o33333333 -6333333 czg o2§ 6? Q3- °'6'56'5 +I°°°°E§ ?353 N423 — 5060.15 0125. —— 61 I 10059335855
-Hoo; 35 34772. Parimo do ex eodcm
    SOZSFQQZ qrdine procedunrralO§
36O§ Ij@ g £10065-?=3?=»%3%1, ?~3-?~.%f, 95' 3 rox 7980 Q; $$l§§ ~Y, $3-33%, §§ €§ ?. Ex fecundo ordin¢ ¢ Z 2 z; 62 566666666; - '6§ 6666 4 K 4 . 64 -64 lo§ 6666 5 If 182357 -|- 20002667306 3* ~ ';'. lQ°?94@ ° 5689 2o2oz7o73I6~;:; “ggi 198016272;'6;-Z; 34i Hand fccu§ ;;.;<;- 63 dem ordine eliciuntur -~'°""'°"'—— rariow 13%, $5552 +2om255464 Tsssii,
    i%{2%'ZEi
U
11- ... 223143551285-1.&; § L;56§ ¢; l0Q°7G ¥zs¢'e"e: g &¢ ° Yi 405465 1°8o8f§ =§ °"Pf- P38456 '?i3 '?9 5 “64 ° ... Q Q 287682.07344:} 3 5 693 1471805z.§ :~;=L2.=ii 8 X 3 62079441 5956, =.- L8fii -1- 5 2302S3509584¥;- E-.fL1@i ¢X P¢~ PQS- S;5 a>1017980%§ 5 1° § Q 2397395'27z64;T:Lurii I1 g 6 10986zz28860;-|-;;:.L3»ii Simi ​Similes ordines ȧ 3$^{rio}$, 4$^{rio}$, & quovis alio numero derivari possunt, suas quisque rationes exhibiturus.
Acquisito Log-o 10$^{rii}$, consicienda est conscienda est statim tabell reducendorum Logorum naturalium ad Tabulares, ut qævis ratio, simul ac inventa est, reducatur ad mensuram tabularium; ita enim Log-i compositorum, quorum ope ad primorum Log-os descenditur, simul sient Tabulares absque reductione.
Fiat igitur, ut Log-us 10$^{rii}$ non-tabularis 1302585, ad tabularem 10000000; ita 1, ad 4,3429448. Hic numerus bis, ter, quater & pluries sumptus constituit tabellam reducendorum Log-orum namralium ad tabulares, quam hic fubjectam vides.
1 043429448190
2 086858896380
3 130288344570
4 173717792761
5 217147240951
6 260576689141
7 304006137332
8 347435585522
9 390865033712
Hujus igitur ope tabellæ, rationis 98/100 mensura naturalis 20202707316 reducitur ad tabularem hoc modo:
2 086858896381
0 0
2 0868588964
0 0
2 08685890
7 3040061
0 0
7 30401
3 1303
1 043
6 26
87739243069
Tum à Log-0 100$^{rii}$ auseratur rationis 98/100 mensura festat unde ablato Log-211 restat —— —— ———
20000000000000
87739243069
19912260756031 = L 98
16901960800291 = L 99
8450980400145 = L 7
cujus semis ——— ——
Item rationis 100/102 mensura naturalis 19802627296 reductam sit 86001717619.
Ergo ja Log-0 100$^{rii}$ adde rationis 100/102 mensurā sit —— ——— — ——
20000000000000
86001717619
20086001717619 = L 102
7781512503836
12304489213783 = L 17
unde ablato Log-0 6$^{rii}$ restat {[bar|3}} ———
Hic tabula numerorum primorum egregium usum præstare potest.
1 1,5 i,5
2 2,25 i,125
3 3,375 i,125
4 5,0625 i,265625
5 7,59375 i,51875
6 11,390625 i,8984375
i5 i125 i125 i265625 i518 i89 22
+ 15114040
− 1137845
16251885⁠17/20⁠
13976195⁠20/23⁠
Cæteriùm isthæc omnia & longè plura ex prop. 13, 15, & 16 Logarithmotechniæ nostræ apertè (illegible text) non magis considerando hyperbolam, quàm si ea nusquam in rerum (illegible text) extitisset. Quare frustra sunt, qui hyperbolam ad constructionem Logarithmorum vel hilum conferre autumant; imo Logarithmorum ope quadrare hyperbolam, verius est. Id quod exemplo ostendere haud pigebit. In diagrammate (Fig. 1.) sit A H 74305816 parium, qualium A I est 1, & oporteat invenire aream B I H F.
Opus est ad eam rem tabella subjecta, quæ continet Log-os naturales suprà acquisitos, in priori, columna ab 1 usque ad 9, in altera à 10 usque ad 1000000000.
1 00000000000 02,30258508299
2 69314718052 04,60517018599
3 109861228860 06,90775527898
4 138629436104 09,21034037198
5 160943791232 11,51292546497
6 179175946912 13,81551055796
7 194591014904 16,11809565096
8 207944154156 18,42068074395
9 219722457720 20,72326583695
Tum prima figura numeri dati semper diftinguatur à sequentibus separatrice, hoc modo: 7,4305816, & ipsi primæ figuræ semper adjiciatur 1, ita conslantur, hoc loco, 8. Quærenda est nunc rationis 8 ad 7,4305816 mensura naturalis. Id ut fiat commodius dic: ut 8 ad 7,4305816; ita 1 ad 0,9288227, hunc quartum proportionalem auser ab 1, reliquum 0,0711773 voco potestatem primam, quæ ducenda est in se ita, ut in facto idem numerus partium extet, qui erat in ipso 0,0711773; productum 0,0050662 est potestas secunda, quæ rursus ducatur in primam 0,0711773, ut idem numerus partium extet, prodit 0,0003606, quæ est tertia potestas; & eodem modo invenitur-quarta 0,0000256, & (illegible text) 0,0000018. Deinde ​Potestas prima 0,0711773
Et secundæ semis 25331
Et tertiæ triens 1202
addantur
Et quartæ quadrans 64
Et quintæ pars quinta 4
summa --- 1,0738374 est mensura rationis 8(illegible text)
7,4305816, eadem scilicet cum ratione 80000000 ad 74305816. Porto Log-us absoluti 80000000 facile acquiritur ex superiori tabella; cum enim index primæ figuræ numeri 80000000 fit 7, è regione 7fii ex secunda columna excerpo Log-um absoluti 10000000 (hoc est unitatis septem cyphris affectæ).
qui reperitur 16,11809565
cui subscribo Log-um 8lii 2,07944154
addo
summa est Log-us absoluti 80000000 = 18,19753719
ablata mensura rationis 80000000 ad 743005816 = 0,0738374
restat Log-us absoluti 74305816 = 18,1236997, atque
tanta est area BIHF.
Mantiffæ loco accipe modum facillimum quadrandi quamvis hypeiebolæ partem per Log-os tabulares. Dati numeri 74305316 Log-us tabulatis est 7,87102278, per superioris tabellæ columnam secundam reducendus ad naturalem, proditque eadem, quæ supia, area BIHF = 18,123699872.
Postremo, ne quis hæsitacioni locus restet, accipe, quo pacto ex Prop. 13, 15, 16. Logarithmot. calculum superiorem derivem.
Differentia terminorum rationem quamvis exprimentium si concipiatur divisa in partes æquales innumeras; composita erit ratio tota extremorum terminorum ex innumeris ratiunculis terminorum à minimo ad maximum infinirissima parte ipsius differentiæ se mutuo excedentium. Sin iidem illi termini innuveri accipiantur pro mediis Arithmeticis aliorum terminorum simili parte infinitissima distantium; summa omnium rariuncularum posterioribus hisce terminis incercedentium deficiet à tota ratione extremorum, non nisi semisse primæ & uliimæ ratiuncularum à prioribus terminis contentarum, id est, ratiuncula minori, quam quæ ullis numeris exprimi posit. Quare posito Maximo termino = 1, & parte infinitissima differentiæ = i, & mensura rationis minimæ itidem i; erit ut medium Arithmeticum terminorum rationis minimam proxime præcedentis, ad medium Arithmeticum terminorum ipsius minimæ; ita mensara minimæ, ad mensuram proxime majoris; hoc est:
1 f- i 1:: i. i~|- ii + i* 'i* écci mci1f'ui'xulrii11aeQ 1— zi. 1:: i . i zii 4i' 8i* &c. penultimate add. 1 —3i . 1:: i . i-}- 3ii-- 9i'-{- 2714 &c. antepenu1:imx.§ ii: summa i'atiu11cu!.:3i~§ f6ii-}-14i*~}- 3,614 écc.: numero terminorum, ​rum, plus summa eorundem terminorum, plus summa quadratorum ab iisdem &c.
Sin minimus terminus ponatur = 1, manentibus carteris ut supra; evadit summa ratiuncularum = 3i — 6ii -|- x4i' — 36%, &c.
Hinc data differentia terminorum = ol; erirnumerus terminorum = o§ & per 16 Logarithmot. summa corundem terminorum = 0, 005, & summa quadratorum = 0, 000333. At data gdilferentia terminorum; olgq; numerus rerminorum est: o, oi, 8; summa eorundcm: 0, 00005, 6: summa quadratorum = 0,00000333, &c.
Nota.Prop. IV. Logarithmot. Signa speciebus intercedentia debebant esse alternatim affirmata & negata: atque ubicunque, Lector offenderit infinitissimam, legat infinitesimam.
Errata.
Page 742. l. 25. put a comma after open'd, (which is material for the sense.) p. 749. l. 16. r. idque. ibid. l 40. r. magnitudinum. p. 753. l. 4. $-a+a^{2}$, $-a^{3}$, p. 754. l. 19. r. Huic. p. 755. l. 11. r. $b^{2}a^{2}+b^{2}a^{3}+b^{2}a^{4}$. ibid. l. 14. r. $a^{2}+a^{3}$. p.756. in Fig. 1. the letters appearing obscure, those, that denote the small lines parallel to the Asymptote NA, are IB. ps. qt. ru. And the other capital letters are GFH. GBA. GMN.
In the SAVOY,
Printed by T.N. for John Martyn, Printer to the Royal Society, and are to be sold at the Bell a little without Temple-Bar, 1668.