Philosophical Transactions/Volume 3/Number 34

Numb. 34:

PHILOSOPHICAL

TRANSACTIONS.

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Monday, April 13. 1668

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The Contents.

The squaring of the Hyperbola by an infinite series of Rational Numbers, together with its Demonstration, by the Right Honourable the Lord Viscount Brouncker.An Extract of a Letter sent from Danzick, touching some Chymical, Medicinal and Anatomical particulars.Two Letters, written by Dr. John Wallis to the Publisher; One, concerning the Variety of the Annual High-Tides in respect to several places: the other, concerning some Mistakes of a Book entituled SPECIMINA MATHEMATICA Francisci Dulaurens, especially touching a certain Probleme, affirm'd to have been proposed by Dr. Wallis to the Mathematicians of all Europe, for a solution.An Account of some Observations concerning the true Time of the Tydes, by Mr. Hen. Philips. An Account of three Books:I. W. SENGWERDIUS PH.D. de Tarantula.II. REGNERI de GRAEF M.D. Epistola de nonnullis circa Partes Genitales Inventis Novis.III. JOHANNIS van HORNE M.D. Observationum suarum circa Partes Genitales in utroque sexu, PRODROMUS.

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The Squaring of the Hyperbola, by an infinite series of Rational Numbers, together with its Demonstration, by that Eminent Mathematician, the Right Honourable the Lord Viscount Brouncker.

WHat the Acute Dr. John Wallis had intimated, some years since, in the Dedication of his Answer to M. Melbomius de proportionibus, vid. That the World one day would learn from the Noble Lord Brounker the Quadrature of the Hyperbole; the Ingenious Reader may see performed in the subjoined operation, which its Excellent Author was now pleased to communicate, as followeth in his own words;

My Method for Squaring the Hyperbola is this:

Let AB be one Asymptote of the Hyperbola EdC; and let AE and BC be parallel to th'other: Let also AE be to BC as 2 to 1; and let the Parallelogram ABDE equal 1. See Fig. 1. And note, that the Letter x every where stands for Multiplication.

Supposing the Reader knows, that EA. αζ. KH. βη. dθ. γχ. δλ. εμ. CB. &c. are in an Harmonic series, or a series reciproca primanorum sue arithmetice proportionalium (otherwise he is referr'd for satisfaction to the 87, 88, 89, 90, 91, 92, 93, 94, 95, prop. Arithm. Infinitor. Wallisij:)

I say	$ABCdEA={\frac {1}{1\times 2}}+{\frac {1}{3\times 4}}+{\frac {1}{5\times 6}}+{\frac {1}{7\times 8}}+{\frac {1}{9\times 10}}$ &c.	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ infinitum.
	$EdCDE={\frac {1}{2\times 3}}+{\frac {1}{4\times 5}}+{\frac {1}{6\times 7}}+{\frac {1}{8\times 9}}+{\frac {1}{10\times 11}}$ &c.
	$EdCyE={\frac {1}{2\times 3\times 4}}+{\frac {1}{4\times 5\times 6}}+{\frac {1}{6\times 7\times 8}}+{\frac {1}{8\times 9\times 10}}$ &c.

For (in Fig. 2, & 3) the Parallelog.		And (in Fig. 4.) the Triangl.	Note.
	$CA={\frac {1}{1\times 2}}$	$EdC={\frac {1}{2\times 3\times 4}}={\frac {\Box dD-\Box dF}{2}}$	${\tfrac {1}{2}}CA=dD+dF$
$dD={\frac {1}{2\times 3}}$	$dF={\frac {1}{3\times 4}}$	$Ebd={\frac {1}{4\times 5\times 6}}={\frac {\Box br-\Box bn}{2}}$	${\tfrac {1}{2}}dD=br+bn$
$br={\frac {1}{4\times 5}}$	$bn={\frac {1}{5\times 6}}$	$dfC={\frac {1}{6\times 7\times 8}}={\frac {\Box fG-\Box fk}{2}}$	${\tfrac {1}{2}}dF=fG+fk$
$fG={\frac {1}{6\times 7}}$	$fk={\frac {1}{7\times 8}}$	$Eab={\frac {1}{8\times 9\times 10}}={\frac {\Box aq-\Box ap}{2}}$	${\tfrac {1}{2}}br=aq+ap$
$aq={\frac {1}{8\times 9}}$	$ap={\frac {1}{9\times 10}}$	$bcd={\frac {1}{10\times 11\times 12}}={\frac {\Box cs-\Box cm}{2}}$	${\tfrac {1}{2}}bn=cs+cm$
$cs={\frac {1}{10\times 11}}$	$cm={\frac {1}{11\times 12}}$	$def={\frac {1}{12\times 13\times 14}}={\frac {\Box et-\Box el}{2}}$	${\tfrac {1}{2}}fG=et+el$
$et={\frac {1}{12\times 13}}$	$el={\frac {1}{13\times 14}}$	$fgC={\frac {1}{14\times 15\times 16}}={\frac {\Box gu-\Box gh}{2}}$	${\tfrac {1}{2}}fk=gu+gh$
$gu={\frac {1}{14\times 15}}$	$gh={\frac {1}{15\times 16}}$	&c.	&c.
&c.	&c.

And that therefore in the first series half the first term is greater than the sum of the two next, and half this sum of the second and third greater than the sum of the four next, and half the sum of those four greater than the sum of the next eight, &c. in infinitum. For 1/2dD = br + bn, but bn > fG, therefore 1/2dD > br + fG, &c. And in the second series half the first term is less then the sum of the two next, and half this sum less then the sum of the four next, &c. in infinitum.

That the frist series are the even terms, viz. the 2^d, 4^th, 6^th, 8^th, 10^th, &c. and the second, the odd, viz. the 1^st, 3^rd, 5^th, 7^th, 9^th, &c. of the following series, viz. ${\tfrac {1}{1\times 2}}+{\tfrac {1}{2\times 3}}+{\tfrac {1}{3\times 4}}+{\tfrac {1}{4\times 5}}+{\tfrac {1}{5\times 6}}+{\tfrac {1}{6\times 7}}+$ &c. in inifinitum = 1. Whereof a being put for the number of terms taken at pleasure, ${\tfrac {1}{a+a}}$ is the last, ${\tfrac {a}{a+1}}$ is the sum of all those terms from the beginning, and ${\tfrac {1}{a+1}}$ the sum of the rest to the end.

That 1/4 of the first terme in the third series is less than the sum of the two next, and a quarter of this sum, less than the sum of the four next, and one fourth of this last sum less than the next eight, I thus demonstrate.

Let a = the 3^d or last number of any term of the first Column, viz. of Divisors,

${\frac {1}{a\times {\underline {a-1}}\times {\underline {a-2}}}}={\frac {1}{a^{3}-3a^{2}+2^{a}}}={\frac {16a^{3}-48a^{2}+56a-24}{16a^{6}-96a^{5}+232a^{4}-288a^{3}+184a^{2}-48a}}=A$

${\frac {1}{{\underline {2a}}\times {\underline {2a-1}}\times {\underline {2a-2}}}}={\frac {1}{8a^{3}-12a^{2}+4^{a}}}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ $={\frac {16a^{3}-48a^{2}+56a-24}{16a^{6}-384a^{5}+880a^{4}-960a^{3}+498a-96}}=B$
${\frac {1}{{\underline {2a-2}}\times {\underline {2a-3}}\times {\underline {2a-4}}}}={\frac {1}{8a^{3}-36a^{2}+52a-24}}$

${\frac {64a^{6}-384a^{5}+928a^{4}-1152a^{3}+736a^{2}-102a}{64a^{6}-384a^{5}+880a^{4}-960a^{2}-96a}}\times {\tfrac {1}{4}}A<B$.

And $48a^{4}-192a^{3}+240a^{2}-96a=$ Excess of the Numerator above Denomin.

But —— The affirm.	$>$	the Negat.	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ if a > 2.
That is, $48a^{3}+240a^{2}$	$>$	$192a^{3}+96a$
Because $a^{3}+5a^{2}$	$>$	$4a^{3}+2a$
$a^{2}+5a$	$>$	$4a^{2}+2$

Therefore $B>{\tfrac {1}{4}}A$.

Therefore 1/4 of any number of A; or Terms, is less than their so many respective B. that is, than twice so many of the next Terms., Quod, &c.

By anyone of which three Series, it is not hard to calculate, as near as you please, these and the like Hyperbolic spaces, whatever be the Rational Proportion of AE to BC. As for Example, when AE is to BC, as 5 to 4. (whereof the Calculation follows after that where the Proportion is, as 2 to 1. and both by the third Series.)

First then when (in Fig.1.) AE. BC:: 2. 1.

2 x 3 x 4)	1. (0.0416666666——	] 0.0416666666
4 x 5 x 6)	1. (0.0083333333——	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ 0.0113095237
6 x 7 x 8)	1. (0.0029761904——
8 x 9 x 10)	1. (0.0013888888——	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ 0.0029019489
10 x 11 x 12)	1. (0.0007575757——
12 x 13 x 14)	1. (0.0004578754——
14 x 15 x 16)	1. (0.0002976190——
16 x 17 x 18)	1. (0.0002042484——	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ 0.0007306482
18 x 19 x 20)	1. (0.0001461988——
20 x 21 x 22)	1. (0.0001082251——
22 x 23 x 24)	1. (0.0000823452——
24 x 25 x 26)	1. (0.0000641026——
26 x 27 x 28)	1. (0.0000508751——
28 x 29 x 30)	1. (0.0000410509——
30 x 31 x 32)	1. (0.0000336021——
32 x 33 x 34)	1. (0.0000278520——	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ 0.0001829939
34 x 35 x 36)	1. (0.0000233426——
36 x 37 x 38)	1. (0.0000197566——
38 x 39 x 40)	1. (0.0000168691——
40 x 41 x 42)	1. (0.0000145180——
42 x 43 x 44)	1. (0.0000025843——
44 x 45 x 46)	1. (0.0000109793——
46 x 47 x 48)	1. (0.0000096361——
48 x 49 x 50)	1. (0.0000085034——
50 x 51 x 52)	1. (0.0000075415——
52 x 53 x 54)	1. (0.0000067193——
54 x 55 x 56)	1. (0.0000060125——
56 x 57 x 58)	1. (0.0000054014——
58 x 59 x 60)	1. (0.0000048704——
60 x 61 x 62)	1. (0.0000044068——
62 x 63 x 64)	1. (0.0000040002——

	0.0416666666
	0.0113095237
	0.0029019589
	0.0007306482
3)	0.0001829939	( 0.0000609980
	0.05679179
+	0.00006100
	0.05685279	< EdCy
But	0.0007306482	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ ${\overset {\cdot \cdot }{\underset {\cdot \cdot }{-}}}$
	0.0001829939
	0.0000458315
Therefore	0.05679179
+	0.00004583
+	0.00001528
	0.05685290	> EdCy.

For, it has been demonstrated that 1/4 of any terme in the left Column is less than the terme next after it; and therefore that, of the last terme, at which you stop, is less than the remaining terms, and that the total of these is less than 4/3 of a third proportional to the two last.

And therefore ABCyE being	=	0.75 ————	——————	0.75
and Ed Cy	>	0.05685279	——— and <	0.05685290
And ABCdE is	<	0.69314720	——— and >	0.69314709

But when AE . BC :: 5. 4. or as EA. to KH. then will the space A B C E. or now, the space AHKE (AH = 1/4AB.) be found as follows.

8 x 9 x 10)	1. (0.0013888888——
16 x 17 x 18)	1. (0.0002042484——	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ 0.0003504472
18 x 19 x 20)	1. (0.0001461988——
32 x 33 x 34)	1. (0.0000278520——	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ 0.0000878204
34 x 35 x 36)	1. (0.0000233426——
36 x 37 x 38)	1. (0.0000197566——
38 x 39 x 40)	1. (0.0000168691——

	0.0003888888
	0.0003504472
3)	0.0000878204	( 0.0000292735
	0.0018271564
+	0.0000292735
	0.0018564299	< Eab
But	0.0003504472	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ ${\overset {\cdot \cdot }{\underset {\cdot \cdot }{-}}}$
	0.0000878204
	0.00002200737
Therefore	0.0018271564
+	0.0000220074
+	0.0000073358
	0.0018564996	> Eab

Therefore E M b. (Fig. 4.)
being	=	0.025————	———	0.025
Eab	>	0.0018564299	——&<	0.0018564996
EMBa(Fig. 4.) or EKM (Fig. 1.)	>	0.02685643—	———<	0.02685050
AHKM	<	0.22314356—	———>	0.22314349

Therefore 3 A B CdE	=	2.07944154
and AHKE	=	0.2231435———
ABCdE (when AE.BC :: 10 . 1.)	=	2.3025850———

Therefore the Logar. of 10
is to the Log. of 2
as 2.302585

to 0.693147

An Extract

Of a Letter, written by the Honourable Consul of Danzick, the Heer Michael Behm, to the Excellent Johannes Hevelius, Consul of the same City (by whom it was communicated to the Publisher) touching some Chymical, Medicinal and Anatomical particulars, here deliver'd in the same Language it was written in.

Nobilissime Dn. Heveli, Amice Sc Collega plurimum honorande,

Perlegi hisce diebus dia desideratam Versionem Latinam Illustr. Domini Boylii tum de Coloribus, tum de Fluiditate & Firmitate. Plurima observavi subtilis ingenii & peritarum in Chymia mamuum Oracula. Utinam Vir Illustris de Natura Salium Experimenta sua ampliora edere dignaretur, quomodo Alcalia sen Lixivialia, utpote fixa, ab aliis Acidis Austerisque, non minus fere fixis, discrepent, & utraque a Volatilibus aliisq; speciebus Salium nondum satis explicatis, atque crasi nomineque distinctis. Nam inter Volatilia diferre videmus Sal Urinosum a Sale, quod inest Spiritui Vini aliisque inflammabilibus; quia commixt a coagulantur, & novam indolem acquirunt: Taceo Mixturas liquoris ex Nitro fixo cum ejusdem Spiritu acido, de quibus bene egerunt Dn. Boyle aliique.

Magnam spem habeo inveniendi liquorem, qui vesicæ injectus blande comminuat Calculos, aliasque mixturas, qua in Ventriculo noxias atque varias visciditates (quas Tartaro plurimi adscribunt immerito) præcaveant vel attinuent. Utinam Dn. Glisson, Dn. Wharton, aliique, ex Anatomicis Chymicisque experimentis: explicare vellent, Lymphæ, Sanguinis ejusqueseri degenerationes, & conglutinationis atque corruptionis modos & causas.Aliquid bac in re non ita dudum tentatum præstitumq, suit (præcunte Clarissimo Dn. Dele Boe Sylvio) a Doctissimo Matthia Paisenio, Hamburgensi, in Disput Medici Inaugurali de Humorum vitiis eorumque Restitutione, habiti & impressa Lugduni Batav.Nuper Serumex sanguine Brutorum examinavi, & calore levinsculo coagulari vidi adinstar fere Albuminis ovi; potissimum addito Acido indurabatur: sed ab admixto spirtu vini serum tenebatur diu liquidum, magis tamen ab Alcalibus. Propediem accuratiora tentabo. Faciebam id, fateor, dum a Podagra vexabar, que me a curis Politicis ad hæc & varia Medicinalia adegit. Notavi sane, Podagram & Arthritidem oriri, ubi Urinasa putrilago non separatur per Renes aut sudores a mass a sanguinea, sed cum ea distribuitur, & circa juncturas in frigidioribus ligamentis hæret, ubi propter salis acredinem dolores acriores (difflabiles tamen) vel propter glutinositatem Tophos aut rigores junctorarum efficit. Utinam hujus mali causas sublimia Anglorum Ingenia penitius investigare & Orbi communicare vellent, ne Incurabile amplius inter Medicos censeatur. Idipsum me valde fecit anxium, præsertim cum diligenter notarem, nil prodesse Purgantia, nil Sanguinis Missiones, parum Sudores & Cauteria, obesse Emplastra oleosa & refrigerania omnia; quinimio, Spiritum Vini & Salis Armoniaci non satis elicere aut discutere totam Materiam. Thermas valde salubres esse, norunt cuncti, & ipse expertus sum, præsertim que Urinam provocant. Sed quia non ubique suppertune Thermæ, prabuit mihi Curiositas Liquorem, qui odore, sapore, virtute satis exacte æmulatur Thermas, ista ratione membra agra, reliquum vero Corpus simplici calida balneando, levamen magnum sentio. Porro expertus su, ausu proprio, insignem medelam per pilulas, qua Urinam promovent, sanguinem depurant, fluidiorgemque reddunt, Caleuli materiam atque Scorbuticas in Mesentterio visciditates referant, absgue alviturbatione. Verum per Vesicatoria (diffuadentibus licet Medicis) dolenti parti adhibita, optimam, promptam atque tutam hactenus per aliquot annos mihi & Amicis inveni medelam; quamvis eam aliis, ad fistulosa ulcera pronis, non suaferim. Ex Magneticis Transplantationibus nullum luculentum scio levamen. Utinam alii suas quoque medelas candide proferrent, non enim omnia profunt omnibus.

Insignem atque felicem in Anatome industriam Dn. Highmori & aliorum veneror, quoties hooris subcifivis salia legere, aut nonnulla amplius per sectiones Avium, Piscium, & Quadrupedum scrutari possum. Usum quippe partium diversa Animalia magis elucidant, atque Veterum errores oftendunt.

Lienem Dn. Highmorus & alii merito absolvunt ab Acido Melancholico, atque sanguificatione. Vidi aliquoties Lienem, calentem adhuc, Pulmonum imitari spongiositatem, & non Aere solum, sed & Coloratis liquoribus valde distendi posse; quo facto magis patent ejus vasa atque connexiones, ususq; a Dn. Highmoro expositi. De eo tamen, pace acutussumi Viri, modeste ambigo, an plurima ista Albicantia vasa sint Nervi spiritu animali turgentes, vel potius Fibre tendinum, dilatationi & contractioni servientes, uti in Pulmonibus. Opinor etenim, Sanguinem, qui in Corde non savis cum Chylo misectur, in Liene tanquam per Spongiam crassiorem colari & misceri plenim, & aquosa panca per Pancreas seponi, & tandem in Hepate iterata colatione Bilem secorni. Et quando per Motum ant vehementes Affectus sanguis in Corde nimis ebullit, tunc, ne ille Cor opprimat, aut nimio impetu Caput infestet, magnam partem recipit Lien, ut ejus intumescentia cum pulfu, Cor emnulante, & calore intenso, a quovis sentiri queat. A vitiis Mesenterii & Hypochondriaco malo, Liemem & Diaphragma infici credo, sed nunquam vel raro ista mala a vitio Lionis provenire putem.

De Morbis, quos ex effervescentia Acidi fucci Pancreatici cum Bile in Duodeno derivat Nobil. Dn. Sylvins, hacetenus valde dubito. Nunquam enim Succum istum sensi acidum Vide, quæ præmemoratus Dn. Paisenius in ead. Disput. hanc in rem differit.; neque Bilem per Acida, sive mediocria sive acria fuerint, effervescere vidi, sed coagulari in fundo, uti ab Acidis solet præcipitari Lac Sulphuris, aliaque Oleosa. Ideo per mixturam Bilis cum Acido Ciborum fermento (quod certius apparet, quam succus iste) Chylum utiliter temperari, cum Helmontio credo. Tot vero Morbos ex mixtura tali, licet faceret effervescentiam, oriri negat experientia. Sed de cunctis hisca Juidcium Regiæ Anglorum Societatis, quam ceu Lumen Seculi humillime veneror, submisse expecto, Gedani, 11 Novem, 1667.

A LETTER

Written by Dr. John Wallis to the Publisher, concerning the Variety of the Annual High-Tydes, as to several places; with respect to his own Hypothesis, deliver'd N°. 16, touching the Flux and the Reflux of the Sea.

SIR, In my Hypothesis for Tydes, you may remember, that I cast the Annual High-Tydes not on the Two Æquinoxes, about the 11. of March and September; nor yet on the Apogæum and Pegæum of the Sun, about the middle of June and December; but (as proceeding from a Complication of those two Causes) on a Midle time between the Perigæum and the two Æquinoxes, (like as is the greatest Inequality of the Natural daies, proceeding from a Complication of the same Causes.) And particularly, for the Coast of Kent (and consequently the Rivers of Thames and Medway) about the beginning of November and February: which agrees with Observations on those Coasts, and particularly with that of yours of Febr. 5. this year.

The last year, when I was present in the R. Society, I remember, an account was brought us of the Annual High-Tydes on the Severn, and at Chepstow-bridge, to be about the beginning of March, and the end of September. Which though they agree not with the particular times on the coast of Kent, yet in the general they agree thus far, That the one is about as much before the one Æquinox, as the other is after the other Æquinox. You now acquaint me with High-Tides about February 22. about the coast of Plimouth, which is later than that of the coast of Kent, but sooner than that on the Severn. And I doubt not but in other parts of the world will be found other Varieties.

The reasons of these Varieties are (as I have formerly signified) to be attributed to the particular Position of those parts, rather than to the general Hypothesis. Of which this, in brief, may serve for some account at present. The General Hypothesis of the Earths diurnal Motion from West to East, would cast that of the Waters, not following so fast, from East to West; which causeth the constant Current within the Tropicks, where the Circles are greatest, west-ward from the Coast of Africa to that of America, (which is also the Cause of the constant Eastern Brize blowing in those parts.) Butthe Sea thus beating on the Coast of America, is cast back as with an Eddy on either hand, and consequently returns from the American shore East-ward towards the Coast of Europe; where, the Parallel Circles to the Æquator being less, and consequently the Diurnal Motion slower, doth not cast the waters so strongly West-wards, as between the Tropicks, and so not strong enough to overcome the Eddy, which it meets with from the other Motion, which gives the Sea a North-Easterly Motion (on these Coasts) as to its usuall course. The Current therefore of our Seas being North-Easterly, we are next to consider, at what times it runs more to the North, and at what more to the East. When it runs most Northerly, it runs up the Irish Sea, and so up the Severn: When most Easterly, it runs streight up the Channel, and so to the Coast of Kent: When between these, it beats against Devonshire and Cornwall, and those parts. We are therefore to consider (as to the Annuall periods) that the Annuall Motion of the Earth in the Zodiack, and the Diurnal in the Æquator, are not precisely in the same direction, but make an Angle of 231/2 deg. at the Æquinoxes; but run, as it were, parallel at the Solstices: And as they be nearer, or further from these points, so is the Inclination varied. Which several directions of Motion, do cause the Compound Motion of both to vary from the East and West more or less, according as the Sun's Position is farther or nearer the Solstices, And therefore, nearer to the Æquinoxes, this Inclination doth cast the Constant current of out Seas more to the North and South; and further from it, more to the East and West, Which is the reason why the Current up the Irish Sea is nearer to the Æquinoxes (at the beginning of March and end of September) and up the Channel or Narrow Seas, farther from it (at the beginning of February and of November:) and against the Coasts of Devonshire and thereabout, at some intermediate time. And thus much I thought fit to signifie upon this occasion. Dat, Oxford the 7, of March An. 1667/8.

Another Letter

Written by the same Hand, concerning some Mistakes, to be found in a Book lately publish'd under the Title of SPECIMINA MATHEMATICA Francisci Du Laurens, especially touching a certain Probleme, affirm'd to have been proposed, by Dr. Wallis, to the Mathematicians of all Europe, to solve it.

ACcepi (V C.) ante quatridnum, quam mihi misisti Fransisci Du Laurens Tracatum, cui titular, SPECIMINA MATHEMATICA, &c. eumq; mox evolvi, quo Tibi possim (quod petis) quid de eo sentiam, paucis offendere. Videtur autem plus fronte polliceri, quam opere absolvit. Prioris libri pars magna, ex Oughtredi meisque scriptis (utut neutrius ibi meminerit) videtur desumpta, idque tam manifeste, ut non modo peculiares loquendi formulas, sed & ipsa symbola Notasque passim retineat. Posterioris, non parum ex Vieta, Schotenio, aliisque ab eo editis (quorum & subinde meminit) desumptum. Occurrunt inibi aliqua parum sana, &, minime accurata multo plura. Quænam autem sint illa Genuina Principia, Veraque Geometriæ Elementa, hucusque nondum tradita, quæ Titulus pollicetur, non reperio: Longeque diversimode Hic & Ego sentimus, dum pag. 141. Neminem esse, opinatur, qui hæc sua non præferat ingenti Euclideorum Elementorum Multitudini.

In calce, manifestam mihi facit injuriam, ea de me affirmans, quæ vera utique non sunt. Appendicem quippe subjungit, cui sepciosum hunc fecit Titulum, Solutio Problematis, à D. Wallisio totius Europæ Mathematicis propositi, sed prius ad generale revocati, A. MDCLIII. eodem tempore, quo propositum erat.

Post Titulum, hæc sequuntur. Problema D. Wailisii, Datis Ellypseos**Pro, Ellipseos, errore Typographi, sine dubio maximis Diametris, tum puncto in transversa ejus Diametro assignato, reperire in numeris segmenta lineæ intra Ellypsim terminatæ, & per datum punctum transeuntis, atque datum angulum cum dicta diametro facientis.

Verum quia propositæ Quæstionis solutio æque facilis est in numeris, ac in lines (ut postea apparebit) melius facturum me judicavi, si prius demonstrationem Analyticam hic afferrem, ex qua tum Numerico, tum Geometrica sequeretur, ad problematis solutionem pertinens, effectio. Atque ut hæc solutio cum fænore detur, speciale D. Wallisii problema ad generale sic revoco.

(Postque hanc Præfationem Problema sequitur tanquam suis verbis expositum cum sua ejusdem solutione per septem continuas paginas.)

Ad quæ hæc dicenda nunc habeo.

1. Totius Europæ Mathematicus. ob rem hujusmodi, in arenam vocare, jastantiæ genus est, cujus ego hact.nus reus non sui (credo) nec futurus.

2. Silibuisset (ostentandi gratia) sic fecisse, ilegissem certe quod vel majoris esset difficultatis, vel majoris momenti, Problema, quam hoc esse videtur, utpote quod mediocris Algebrista, primo intuitu, semibore spacio facile solveves.

3. Nec sane hoc Problema, nec quod huic æquipolicat, unquam Ego (quod memini) ulli mortalium, nedum totius Europæ Mathematicis, preposui (nescio an ulli unquam propositurus:) nec quicquam bujus, quod de me persuaderi sibi passus est, verum est.

4. Erat quidem aliquando Problema huic non prorsus absimile mihi propositum (cujus & solutioneni protinus expediebam) sed à me propositum nemini, quod quatenus me spectare possit, videas in Epistola quadam mea, ad Nobiliss Vice-Comitem Brounker data Maij 11. 1658. (quen annum inuuit D. Dulaurens) eoque anno in meo Commercio Epistolico p. 171. typis vulgata, in hæc verba;

Sub initium Februarii jam proxime elapsi, amicorum non-nemo, cui forte occurrebam sero vesperi, quæstionem sequentem mihi porrexit in scriptis, quam jam nuperrime intelligo typis vulgatam esse cum hac Epigraphe; "Spectatissimos viros, Matheseos Professores, & alios præclaros in Anglia “ Mathematicos, ut Problema solvere dignentur, Jean de Montfort maxi- “ me desiderat.

“ Extremis Ellipseos Diametris, distantiâ centri ab aliquo puncto in Axi “ transverso, ubi linea eundem secet sub angulo dayo, in numeris datis; " segmenta ejusdem linæ (siopus est) productæ, & intra transversum Axem “ & Ellipsin terminata, in numeris invenire.

Hanc Ego quæstionem, suam ratus (neque emim vel innuebat Ille, vel Ego tum seiscitabar cujus erat,) paulo adhuc universalius expositam, sub ea sere, quæ subesi, forma (neque enim ipsissima verba memini) postero mane solvebam: Nec eram de illa ultra solicitus (quippe quæ nec magnæ videbatur difficultatis, nec momenti,) quam etiam, ut nunc audio, varii variis modis solvebant, ut ut eurom solutiones nondum viderim.

(Ac deinde sequitur mea istius Problematis, universalius adhuc prepositi, solutio, cum annexa demonstratione, brevis & perspicua; saletem si excipias præli sphalmata; quæ tamen qui hæc intelligit, facile restituet.)

Atque hoc omne illud est, quad Ego de hoc Problemate fecerim, quo id ullatenus ad me spectare videatur. Num autem hoc sit (quod vult D. Dulaurens) Problema illud totis Europæ Mathematicis, a Me propositum esse; Ego cuilibet judicandum permitto, qui Latina intelligit, utcunque fuerit Matheseos ignarus.

Qui illud mihi monstrabat Problema (scriptum primò, postea typis impressam) est Dr. Richardus Rawlison. Quis autem fuerit ille Jean de Monfort qui proposuit, ignoro. Hujus autem impressa Chartula Londini tum temporis satis prostabat pluribusque Matheseos perîtis proposita cen Problema exhibens ex Galia delatum, quod & ex nostris aliquot Londini solvebant: Quorum unus (Dr. Christoph. Wren, tunc quidem in Collegio Greshamensi Londini,nunc,, Oxonii Astronomiæ Professor) solutionem suam typius editam publici juris fecit: simulque (in eadem charta) reposuit Problema aliud, quod ipse præstantissimis in Gallia Mathematicis (uti illud inde nobis in Anglia solvendum proposuit; quod illorum nemo (quod sciam) hactenus solutum dedit. sua verò istius solutio extat in meo de Cycloide Tractatu. p. 72. 73.

Cum itaquq sint hæc omnia (quod dicitur) publica & notoria, non possum non mivari, quo animo D. Dulaurens palàm & in publicum ederet rem tam ab omni veritate alienam.

A Letter written to Dr. John Wallis by Mr. Henry Philips, containing his observations about the True Time of the Tides.

WOrthy Sir, Being desired by Mr. O. to give in, what informations I could, concerning the Tides, I have made bold to present this Paper to your Consideration; which though it have little or no relation to your more curious Philosophical Experiments, yet, I hope, will be of very good use for the finding out the True time of the Tides at all times of the Moone, which is (I conceive) of as great concernment, as any thing in the Motion of the Tides.

For, this time of the Tides, though it be a very necessary thing to be known, yet is very rudely and slightly reckoned up by most Seamen and Astronomers; most of them reckoning, as if the Moone being upon such a set point of the Compasse (as the Seaman calls it) or so many houres past the Meridian (as the Almanack-Makers reckon) it were High-Tide in such and such a Port at all times of the Moone. And thus they reckon the Tides every day to differ constantly 48 m. As for instance; A South-West Moone makes a full Tide at London, that must be understood, that it is High-Tide at London when the Moon is three hours past the Meridian. Now this is true indeed at the New and Full Moon, but not at other times of the Moone, which few take any notice of: only Mr. Booker had wont to give this Caveat, that about the first and last quarters of the Moone, the Neap-tides did not flow so long as the Spring-tides by one point of the Compasse; but he gives no rule to proportion the difference.

But observing this more narrowly, I find, that at London the Tides fall out at the least two points, that is, one hour and an half sooner, in the Quarters then in the New and Full Moone. Now this being a very considerable difference of time, which might very well make many Seamen and Passengers to lose their Tides, I set my self to watch this difference of the time of the Tides, and to find out some Rule, how to proportion the time of the Tides between the Spring-tides and the Neap-tides, and I found by many trialls, that the true time of the Tides might be found out to be somewhat shorter and shorter, from the New and Full Moone unto the Quarters; yet not in an equall manner, neither gradually decreasing from the New and Full Moone untill the Quarters; but rather, that there was some little difference of alteration both at the New and Full Moones, and also at the Quarters; and that the greatest difference fell out in the midst between them, agreeing very well to a Circular proportion, after this manner: (See Fig. 5.)

First, Divide a Circle into 12 Equal parts, or hours, according to the Moones motion or distance from the Sun, from the New Moone to the Full.

Secondly, Let the Diameter of the Circle be divided into 90 parts or min. that is, according to the time of the difference of Tides between the New or Full Moone, and the Quarters, which is one hour and an halfe.

Thirdly, Make perpendicular lines cross the Diameter of the Circle, from hour to hour.

Fourthly, Reckon the time of the Moones coming to the South in the circumference of the Circle, and observe the Perpendicular-Line, that falls from that point upon the Diameter; and the proportionall Minutes, cut thereby, will shew, how many Houres, or Minutes are to be subtracted from the time of High-tides at the New and Full Moone, that so you may have the true time of the Tides that present day.

For Example; At London, on the day of New and Full Moone, it is High-Tide at London at 3 of the Clock, that is, when the Moone is three hours past the Meridian: and so by the Common Rule, the, Moone being about four dayes old, it will be South about three of the Clock, and it will be High-tide three houres afterwards, that is, at 6 of the Clock. But now by this Rule, if you count this time of the Moones coming to the South in the Circumference, the perpendicular-line, which comes from 3 to 9, cuts the Diameter in the halfe, at 45 min, which shews, that so much is to be abated from the time of High-tide in the New and Full Moones; So that it is High-tide 45 min. before 6 of the Clocke, that is, at 5. hours 15 min. and not at 6 of the clock, according to the common-Rule.

The like you may do for any other Port or place, knowing the time of High-water at the New and Full Moon in that place: And you may do it the more readily, it you set down the time of High-water at the New and Full Moon under the Diameter, as I have done for London, where it is high-tide at III. of the clock, So that when the Moon is South at III. of the clock, the perpendicular cuts the diameter at Il. hours 15. m. which added to the time of the Southing makes it V. hours 15. m. and so when the Moon is South at IX. of the clock, by adding 2 h. 15 m. you have the time of high-water, which is XI. of the clock and 15 m.

And thus you may easily make a Table, which by the Southing of the Moon, shall readily tell you the time of High-tide at any time of the Moon, as I have done here for London: To which all other places may be reduced to correspond.

Moon South		Tide London			Moon South		Tide London			Moon South		Tide London			Moon South		Tide London
H.	M.	H.	M.		H.	M.	H.	M.		H.	M.	H.	M.		H.	M.	H.	M.
XII	0	3	0		III	0	5	15		VI	0	7	30		IX	0	11	15
	10	3	9			10	5	21			10	7	41			10	11	29
	20	3	18			20	5	27			20	7	52			20	11	43
	30	3	27			30	5	33			30	8	4			30	11	57
	40	3	36			40	5	40			40	8	14			40	12	10
	50	3	45			50	5	46			50	8	25			50	12	24
I	0	3	54		IV	0	5	52		VII	0	8	36		X	0	12	37
	10	4	2			10	5	59			10	8	48			10	12	50
	20	4	9			20	6	6			20	9	0			20	1	3
	30	4	16			30	6	13			30	9	13			30	1	16
	40	4	23			40	6	20			40	9	26			40	1	29
	50	4	30			50	6	18			50	9	39			50	1	42
II	0	4	37		V	0	6	36		VIII	0	9	53		XI	0	1	54
	10	4	44			10	6	44			10	10	6			10	2	3
	20	4	50			20	6	53			20	10	20			20	2	16
	30	4	57			30	7	2			30	10	33			30	2	27
	40	5	3			40	7	11			40	10	47			40	2	38
	50	5	9			50	7	20			50	11	1			50	2	49

A Table of the time of High Tide for this present year Anno 1668. London-Bridge: where M. denotes Morning, and P. Afternoon.
	Jan.		Feb.		Mar.		April		May		June		July		Aug.		Sept.		Octo.		Nov.		Dec.
	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.	H.	M.
1	12	??	2 P	13	1 P	49	4 P	6	3 P	23	4	18	4	30	5 P	5	5	54	6	36	8	18	8	36
2	1 P	2?	3	11	2	45	3	57	4	4	4	51	4	59	5	32	6	37	7	27	9	20	9	33
3	2	3?	3	56	3	29	4	22	4	39	5	20	5	25	6	2	7	23	8	32	10	24	10	32
4	3	37	4	30	4	8	4	35	5	11	5	48	5	52	6	39	8	28	9	43	11	30	11	36
5	4		5	1	4	40	5	24	5	40	6	18	6	23	7	25	9	41	10	54	12	31	12	46
6	4	57	5	29	5	9	5	55	6	12	6	54	7	1	8	28	11	2	12	3	M	31	M	40
7	5	26	5	58	5	38	6	30	6	49	7	37	7	47	9	43	12	24	M	3	1	32	1	43
8	5	55	6	30	6	9	7	13	7	32	8	27	8	46	11	7	M	24	1	10	2	27	2	37
9	6	2?	7	11	6	49	8	5	8	21	9	27	9	57	12	30	1	35	2	9	3	18	3	28
10	7	5	7	59	7	32	8	58	9	18	10	39	11	19	M	30	2	34	3	1	4	2	4	10
11	7	47	8	53	8	28	9	58	10	19	11	53	12	46	1	48	3	24	3	45	4	37	4	41
12	8	31	9	58	9	28	11	2	11	26	1	16	M	46	2	53	4	6	4	23	5	8	5	10
13	9	33	11	7	10	33	12	6	12	34	1 M	16	2	6	3	45	4	39	4	57	5	37	5	37
14	10	19	12	16	11	40	M	6	M.	34	2	27	3	15	4	24	5	11	5	27	6	6	6	4
15	11	47	M	16	12	44	1	10	1	44	3	38	4	4	4	56	5	41	5	59	6	42	6	35
16	12	54	1	18	M	44	2	13	2	50	4	22	4	41	5	25	6	14	8	34	7	21	7	12
17	M	54	2	18	1	44	3	99	3	49	5	1	5	5	5	54	6	54	7	16	8	5	7	58
18	1	26	3	8	2	37	4	3	4	36	5	24	5	43	6	29	7	45	8	6	8	56	8	49
19	2	50	3	49	3	27	4	45	5	17	6	8	6	15	7	11	8	34	8	57	9	54	9	55
20	3	36	4	24	4	13	5	23	5	53	6	44	6	53	8	4	9	35	9	54	10	58	11	9
21	4	13	4	57	4	50	6	3	6	33	7	25	7	35	9	0	10	40	10	55	12	9	12	30
22	4	44	5	30	5	26	6	48	7	17	8	16	8	28	10	4	11	43	11	58	1 P	17	1 P	52
23	5	13	6	5	6	4	7	42	8	8	9	7	9	27	11	1?	12	44	12	59	2	29	2	2
24	5	41	6	48	6	51	8	38	9	9	10	10	10	33	12	16	1 P	42	2 P	1	3	27	3	58
25	6	13	7	46	7	48	9	35	9	58	11	12	11	32	1P	16	2	34	2	58	4	17	4	41
26	6	5	8	53	8	53	10	44	10	58	12	18	12	48	2	12	3	32	3	48	4	59	5	15
27	7	49	10	10	10	2	11	47	12	1	1 P	21	1 P	49	3	1	4	4	4	32	5	36	5	47
28	8	55	11	28	11	12	12	49	1 P	2	2	19	2	43	3	43	4	41	5	11	6	14	6	19
29	10	16	12	43	12	20	1	47	2	2	3	11	3	27	4	18	5	18	5	49	6	55	7	0
30	11	44			1 P	21	2	28	2	55	3	54	4	16	4	30	5	52	6	31	7	43	7	45
31	1 P	2			2	16			3	40			4	37	5	21			7	21			8	26

These things I have found to fall out right at London for many years, and so I suppose they may in other places. If the difference be not so much between the Neap-tides and Spring-tides in other places, the Diameter must be divided into fewer parts.

As for the higest Tides to happen two or three dayes after the full Moon, I have not made much observation of it, and see little reason for it, but the time thereof agrees herewith. And high Spring-tides are not alwayes alike; this year I have not observed any. I should be glad to hear, how these rules hold in other places, that so this true time of the Tides may be more punctually known.

An Account of some Books.

I. W. SENGVVERDIUS P.D. de TARANTULA, In quo, præter ejus descriptionem, effectus veneni Tarantulæ, qui hactenus fuère occultis qualitatibus adscripti, rationibus naturalibus deducuntur & illustrantur., Lugd. Batav. 1668, in 12.

THis Author having described the structure of the body of this Tarentin Spider, and the chief parts thereof, together with its bigness, qualities, food, abode, manner, and season of stinging, and aptitude to live very many days without any visible food; passeth to discourse of the nature and effect of the poison, which being viscous and tenacious, exserts it self not presenty, but lurks a good while in the body, and after the revolution of a year, being stirr'd and subtilized by the heat of the Sun, is rowsed and put into motion, producing for the first two years only various diseases in the patient, as a dejection of the appetite, burning Feavers, Cachexy; after which do follow very strange and surprizing effects, in some singing and laughing; in others, weeping and crying; in others, sleeping; in others, continual watching; in some, vomiting; in some, dancing and sweating; in some, madness; in some, the fancy of being Kings; in others, that of being slaves: Just as Drunkeness renders some morose, silent and fearful; others bold and clamorous. This only he affirms to be common to all that are bitten by the Tarantula's, that they delight in Musick, and are thereby moved either to dance, or to gesticulate. He observes also, that some of these Patients are delighted with certain sorts of Colours, some with Yellow; some Green; some Red. But he noteth this as the most wonderful effect, that this poison so fixeth those imaginations, which a man chanceth to have when he is bitten, that he, that shall then think himself a King, will persist in that fancy ever after, till he be cured.

The cure of the poisonous effect, he with others assigneth to be Musick, and the dancing, consequent thereto; which the Patients do perform as if taught: Yet so, that not every one is afected with every song, but each with such an one, as is suitable to his temper; that which is unsuitable, tormenting the party,

Some Tarantula's he affirms to have poison contrary to that of others, so that one man bitten by both such, cannot be made to dance; forasmuch as the one poison which induceth to dance, is made ineffectual by the contrariety of the other: It being also requisite, to use different Songs and Instruments, according to the different quality of the poyson, and the various Constitution of the Patient.

Further, he observes, that the Tune, which is suitable to the person bitten, is also suitable to the Tarantula it self, & é contra; as also, that not only men, but other Animals, as Cocks, Wasps, &c. bitten by the Spiders, do dance; and that the effects of the wound depend not on the life of the wounding Spider.

After this, our Author Philosophiseth thus. He esteems, that this poison invades chiefly the Arteries and Nerves, and the Spirits in the Nerves; as also the Brain and Fibres; and having cast the humours, raised in the whole Body, upon the Brain, disturbs the Imagination, according to the diversity both of Men's tempers, and of the Tarantula's.

The Feavers, Cachexy, Dancing, Sleep, Waking, various gestures, he giveth this account of; That since the Tarantula wounds by biting with a moist mouth in the hottest Season; thence by an opening made, the poison, consisting in a Salivous moisture, is out of the Spiders body conveyed into that of Man, whereby, upon its diffusion through the humane body, it affects the Muscles and Nerves principally, and in them, by the periodical heat, the spirits stirring up and increasing the heat in the Heart, and by too vehement heat corrupting the bile in the vessels, and so causing hot Feavers and Cachexy: And it may be also, that by that Salivous and venomous substance in the Mouth, the Pores are obstructed, and the passages shut, wherby an outlet being denyed to the heat, it is too much augmented in the body, and so likewise putrifies the bile, and causeth the same distempers.

Sometime this poison is joyned to the Spirits, and thereby causeth about the beginning of the Nerves another motion, than Nature, if not disturbed, would produce: whence, by vellicating the Muscles, it induces the party to dance. Or it ascends to the Brain, and there, by its viscosity, obstructs the Nerves in the place where they meet, and so hindring the Animal Spirits to pals into the Organs, causeth sleep; or, by its activity (so quick, that the Nerves cannot be obstructed by the Vapors, and a passage is constantly open'd for the Spirits to issue into the Organs) produceth continual Wakes.

If it be demanded, how Musick becomes a remedy, and inciteth the Patient to dance? 'Tis here answered, That Sound having a great influence upon the Actions of Man, and being a motion of the Air, the Air mov'd, causeth a like motion in the next Air, and so on, till the like be produced in the Spirits of the Body, to which the Air is impelled. Wherefore since the commotion of the Passions depends on the Spirits, and the viscous humour of the Tarantula, is a capable subject of sound, hence 'tis (faith this Author) that the Air being mov'd by a Musical Air, suitable to the Patient, the poison of this Spider and the Spirits of the Man are by the same agitation put into a commotion; whence follows a propension to dance: And the Nerves being by the same agitation vellicated, and the Spirits in the Nerves stirr'd more vehemently, and consequently the Muscles moved, the whole Body cannot but perform that dancing motion.

If it be asked, how the cure is made by Dancing, The answer here is, that by that vehement Motion the Bloud is heated, the Pores open'd, the Poison rarified and dispersed, and by Sweat ejected: But that these Patients are not cured by Sudorificks, the reason thereof is given from the difference between Sweat caused by Dancing, and that which is provoked by Medicines, forasmuch as Medicines are not capable so to stirr the little partcles, wherein the poison lodges, as Dancing is.

But if it be insisted, why all that are thus bitten, are not curable by Dancing, some being known to have Danced 30 or 40 years, without being cured; Here is no other cause alledged, but that in some the poison is pertinacious and unrarifiable.

These, and the like Phænomena (viz. why several Patients are cured, and several Tarantulas affected by several Tunes; why the Tarantulas in Apulia only produce these effects, &c.) are fully explicated in the Book it self.

II. REGNERI De GRAAF, M.D., EPISTOLA, De nonnullis circa Partes Genitales Inventis Novis.Lugduni Batlv. in 16, A. 1668.

III. JOHANNIS Van HORNE, M.D. Observationum suarum circa Partes Genitales in utroque sexu, PRODROMUS. Lugd. Batav. in 16, A. 1668.

It seemes, that the two Authors in these two Papers have met with almost the very same Observations; which they account New, about the Genitals in both Sexes; and that, the former having appeared in Print before the latter, the latter thought himself obliged, the self same day that the Epistle of De Graaf came out of the Press (as himself intimateth) to declare in this his Podromus, that, though he knowes not, whether the Observations of the former be altogether the same with his, yet to avoid dispute hereafter, he thought fit, in this Paper of his, to represent the short of his own un-borrowed Observations, concerning that Subject, till he should be able to publish a full history of the structure of those parts.

Touching De Graaf, he 1. Rejects the opinion of those, that teach a Conjunction of the Seminal Arteries with the Veins by visible Anastomoses, and that reckon the Testicles among Glanduls. 2. He affirms, that he hath often unravell'd totam substantiam testiculorum in ingenitem longitudinem. 3 He asserts to have shewd by a short way, Vesicularum feminalium cum vasis deferentibus communionem, magnitudinem, figuram, earumque in Urethram exitum. To which he faith to have added a very easie way of examining the Body of the Prostate. From the consideration of all which he concludes Unam esse solummodo materiam feminis, eamque in testibus generari, in vesiculis excipi, & inde in Urethram ejici, non per unum, ut vult Veslingius, sed duo foramina. 4. He affirmes to have an easier and more accurate way of dissecting the Penis than any other Anatomists he knows; and that he assignes to the Muscles thereof a farr other use, than hath been done hitherto. Of all which he intimateth, that he is ready to publish a Book, after he shall have received the thoughts of Dr. Sylvius: upon the Manuscript thereof. He concludes, that he hath contrived a New Instrument, whereby every one may give himself a Ciyster without any Denudation of the parts, or change of posture.

Concerning Van Horne, he also refutes the above-mention'd Anastomoses between Arteries and Veins; then describes the Spermatick Arteries and Veins; the Pyramidal-Figure, they make, where they meet near the Testes, the direct and retrograde passage of the said Artery's through the Testes, and such a strange Anastomosis between the Spermatick Veines, that they represent a kind of rete mirabile most elegantly. He also will not admit the Testes to be Glandular, but affirms, (which is the same with the Doctrine of De Graaf) totam Testium molem nil esse aliud, quam congeriem minutissimorum funiculorum, habentium seriem continuatam, atque concavorum, pro seminis materia de hevenda: adding, that if the greater Globe of the Epididymis be well examin'd, there will appear through its Membrane such anfractus and funiculorum gyri, as resemble those of the Brain. He holds triplicem materiam seminis, unam, quæ venit à Testibus; alteram, quæ à Vesiculis; tertiam, quæ ex prostatis in Urethram propellitur. He deduceth from the wonderfulness of the Structure of the Penis, Tensionem ejus, & impetuosam seminis pereundem ejaculationem.

After this, he intimates briefly the Observables in Partibus Genitalibus Mulierum, and among other things remarks (what was lately also noted out of Steno's Myologia, Numb. 32. p. 628) Mulicrum testes esse Ovario in Oviparis analogos, they containing perfect Eggs, full of Liquor, and encompassed with a skin of their own, whereof he affirms to have yet some by him, &c.

ERRATA.

NUmb.32.p.617.l. i. read Rotation for Relation. p.624. l.27.r. Herniarum instat p. 625. l.35. r. Angles.

Numb. 33. p.641. l. 32. r. converging, whereas some Copies have conveying. p.642. l. 17. r. I L M K

In this Numb. p.647. l.5.r. the second, for first term ib. l. 20. r. A = B, for A < B.

---

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, 1667.

⁠