An Accompt of Some Books

An Accomp of some Books. 1. Lectiones 18, Cantabrigiæ in Scholis publicis habita, in quibus OPTICORVM Phænomena Genuine rationes investigantur & exponuntur, ab Haaco Barrow, Coll.S.Trin.Socio, Mathef.Profeß.Lucasiano,& Soc.Regiae Sodali. Londini, 1669. in 4°. Impensis Joh.Dunsmore ad Insigne Rose & Coronae, in Cæmeterio D. Pauli. A Brief summary of these Optick Lectures the Excellent Author doth himself deliver in an Epistle to a Friend, who desired such an one from him; when he saith, that in them he endavoureth to promote that part of Opticks, which he undertakes to treat upon, first by explaining and establishing its Principles; then by deriving from them some useful Conjectaries, serving to the explication of the Phænomena: in the mean while attempting to correct some common errours, and to supply some principal defects therein. Towards these purposes, he first examineth the received Hypotheses of this Science, shewing how they should be understood, and how far admitted; assigning also some Physical causes of them, (in the performance whereof he declares himself not to confide much, but only requires the Hypotheses themselves, as not disagreeable to Reason, and very congruous to Experience, to be admitted.) Having first settled the Hypotheses, he first draweth from them some general Corollaries, partly before acknowledged by others, and partly observed by himself; all confirmed by his own demonstrations. Then proceeding to more Special matters, he prosecutes distinctly Catoptricks and Dioptricks, both Plain and Spherical. And slightly passing over Plain Catoptricks (as easy and commonly treated on, truly enough,) he more largely insists upon Spherical Catoptricks, propounding such Theorems, whereby the Intersections, and the Limits of Reflected rays are declared, and together the Appearing places, or images of points radiating both from a great (and, as it were, immense) distance, and from a distance sensibly near, are determined, both in respect of an eye placed in the Axis of the radiation, and in respect to one placed without that Axis anywhere; the which particulars particulars he had not observed in any Book extant to be truly and diligently handled. Then he from the foundation raiseth a Theory concerning both Plain and Spherical Dioptricks, assuming for a ground that rule concerning the Measure of Refractions, which Mr. Descartes first discovered, and which now most of the best writers do admit, and which he judgeth agreeable to truth (for that all deductions from it very well suit to experience) which yet he knows not that any writers have applyed to this purpose, so as to have raised any competent superstructure thereon. And here orderly (first in respect to Plain, then to Spherical Surfaces,) considering points radiating from a distance so great that their rays may be supposed to fall parallel to one another upon the refracting surface, he propounds some Theorems, from which the chief Symptoms of refracted rays do result; their Interjections and Limits are easily discernable; the appearing places or images of such points are defined, both in regard to an eye situated in the Axis of the radiation, and in regard to one placed elsewhere. Then he prosecutes in like manner the same things in respect to Points, radiating from a distance sensibly finite or near. And that the Use of these things may be more ready, and serviceable to practise, he subjoyneth distinctly and particularly the determinations of the places in the Axis of the Images of points however radiating through each kind of lenses. Having dispatched these matters, he toucheth generally concerning the making a judgment about the Appearances of Magnitudes (as to their situation and figure,) which follow those sorts of reflection and refraction: Afterward more specially and copiously, he shews what kind of Images plain objects do yeild from such reflections, and how they may be delineated. Among these things there are interspersed some considerations about divers incident matters, as about the nature of Light, and the causes of different Colours about the Rainbow, or colours appearing in pellucid globes; about some appearances in the vitreous Prism; concerning the linea refractaria, concerning the resolution of problems by Appropriate lines; concerning the properties of the Conical sections in the reflection of lucid rays, &c. II. Lectiones 13. GEOMETRICAe, in quibus (praefertim) Generalia LINEARUM CURVARUM Symptomata declarantur ab eodem Isaaco Barrow, &c. Impensis ejusdem Bibliopolae, in 4°. Concerning the Geometrical Lectures, arguing great depth in the Mathematical Learning, In the first of them, in order to what follows afterward concerning the Generation of Magnitudes, the Deserving Author treateth about the Nature of Time, as it may be considered in Mathematical suppositions about such Generations. In the second are declared the Mathematical Hypotheses about simple Motions (progressive and Circular,) which serve to the Production of Magnitudes, together with some general remarks, about the natures, dimensions, and properties (consequent on such productions) of Magnitudes. There is also a touch about the Method of Indivisibles, explicating, how in some cases it is to be understood and applied. The third treats about the Generation of Magnitudes by composition and concourse of motions. In the fourth and fifth, from one generation propounded of Curve lines, (supposing them produced by two motions, one uniform, the other accelerated) divers Theorems are inferred, implying so many general properties of Curve lines. The next five Lectures do contain many Theorems, and Problems about readily determining the Tangents of Curve Lines, immediately by them, without other computation; particularly, there are divers single Theorems, whereby the Tangents of all Curves commonly known or considered in Geometry (the Conical Sections, Conchoids, Cissoids, Spirals, Quadratrices, &c.) are determined in ways so general, as to comprehend also the like determination of Tangents in regard to innumerable other Curves, generated in a common manner with them. Of those five Lectures the two first are Lemmatical, or preparatory to the Demonstration of the propositions delivered in the other three; wherein yet there be some Theorems not unpleasant. In the tenth is delivered a general Analytical method of determining Tangents, extending to all sorts of Curve lines, both Geometrical and Mechanical (as Monsieur Des-Cartes distinguisheth.) The eleventh Lecture containeth several general Theorems, about about the Dimension of Magnitudes, or the comparison of them with one another. To that there is subjoyned an Appendix concerning the dimension of Circular, and Hyperbolical segments, with divers Theorems and Rules serving to that purpose. The twelfth Lecture containeth also several Theorems concerning the Dimension of Magnitudes, but chiefly respecting the dimension of Surfaces produced by the Rotation of Curve lines, and the dimension of Curve lines themselves. To this Lecture there are also three Additaments; the first whereof containeth some Theorems about the dimensions of Spaces constituted by the Tangents and Secants of a Circle. The second shews how the foregoing Theorems may be demonstrated by the Apagogick way, or by reduction ad absurdum; together with a way of finding the dimension of the surfaces of Conical bodies. In the third divers Problems and Theorems are added, of kin to those of the eleventh and twelfth Lecture. The thirteenth Lecture propounds an Explication of the nature, and constitution of Equations, together with the variety of Roots, their Limits, &c. by construction, and consideration of certain curve lines appropriate to each Equation: with some notes respecting each particularly, and all in general. So much of these two Excellent Treatises: Since the Publication of which, their worthy Author hath been pleased to communicate to a Friend of his some Corollaries, belonging to the second Problem of his third Appendix to the twelfth Lecture; which because we conceive they will be very acceptable to the Mathematical Reader, we shall here, by the Author's good leave, subjoin them in the same Language, he hath written them in, viz. Caeterum, inquit vice, animadverti, potuisse secundo Appendiculae tertiae Lettionis XII. Problemati Corollaria quaedam adponi non injucunda; qualium adscribam unum & alterum. Probl. I. Detur linea quaepiam AMB (cujus axis AD, basis DB,) curva ANE designetur talis, ut ducta libere recta MNG ad BD parallela, qua ipsam ANE sectet in N; sit curva AN aequalis ipsi GM. Curva ANE talis sit, ut si MT curvam AMB, & NS curvam ANE tangant, sit SG. GN :: TG. √GMq — TGq; ipsa ANE proposito faciet satis. Probl. 2. Idem quaad caera suppositis, & constitutis, curva ANE: ANE jam talis esse debet, ut curva AN semper aequetur interceptae rectae NM. Curva ANE jam talis sit, ut sit SG. GN :: 2TG × GM. CMq — Gq; erit ANE curva, qua desideratur. V.fig.eandem. Probl. 3. Datur curva quæpiam DXX, cujus axis DA; reperiatur curva AMB proprietate talis, ut si libere ducatur recta GX ad ipsam AD perpendicularis, ponaturque, SMT curvam AM tangere; sit MS aequalis ipsi GX. Liquet rationem TG ad TM (hoc est, rationem GD ad MS, vel GX,) dari; adeoque rationem TG ad GM quoque dari. Inervit hoc superficiebus designandis, quarum in promptu sit dimensio. Etenim (duet ME ad AD parallelà) superficies solidi, ex Planis BME circa axem DB rotato progeniti, adequat periph. GDX; ut habetur in IIa Lectionis XII. In Leq.XI. Appendice numero 33, de Cycloide profertur Theoremata quoddam; id quod ex hujusmodi generaliori Theoremate deduci potuisse. Sit AMB curva qualibet, cujus axis AD, basis DB; sit item curva ANE talis, ut si arbitrariè ducatur PMN ad DBE parallela, positóque, rectam TN curvam ANE tangere, sit TN parallela subtenæ AM; completo rectangulo ADEG, erit spatium trilineum AEG æquale segmento ADB. Huic suppar theorema tale est: Idem positis; si tam segmentum ADB, quam spatium AEG circa axem convertantur; erit productum segmento ADB solidum producti ex AEG duplum. V. fig.eandem. Et Tangentium porro contemplatione suborta est methodus, per quam expeditissime plurima circa maximas quantitates Theoremata deducuntur; qua ceriè, si tempestivè se objecissent, digna censuisset qua Lectionibus infererentur: ex his indigitabo nonnulla. Sit curva quæpiam ALB, cujus axis AD, basis DB, et huic parallela V.tab.I.n.XIII. LG, λγ; item LT curvam tangat: Theor. 1. Sit m numerus quicunque potestates exponens; si ponatur DGm−1 × TG = GLm; erit DGm + GLm maximum, seu majus quam Dγm + γλm. Th. 2. Idem sumpto numero m, si ponatur BLm−1 × TL = GLm; erit GLm + BLm maximum, seu majus quam γλm + Bλm. Th. 3. Sint numeri quilibet m,n; si ponatur m × TG = n × DG, erit DGm × GLn maximum, seu majus quam Dγm × γλn. Th. 4. Quòd si ponatur m × TL = n × arc BL, erit GLm × BLm maximum, seu majus quam γλn × Bλm. Th. 5. Th. 5. Si fuerit $TG \times GL = DGLB$, erit $DGLB \times GL$ maximum, seu majus quam $D \times B \times \gamma$. Th. 6. Sin $TG \times GL = 2 DGLB$, erit $GL \times \sqrt{DGLB}$ maximum, seu majus quam $\gamma \times \sqrt{D} \times B$. Haud difficili negotio cum hac demonstrantur, tum ejusmodi complura deprehenduntur. III. A Continuation of the MEMOIRES of M. Bernier concerning the Empire of the G. Mogol. English'd out of French. London, Printed for Moses Pitt in Little Brittain, 1671, in 8o. The first Volume of these Memoires, lately also printed in English for the same Bookseller, containing almost nothing but political affairs, was left un-mention'd in these Books; but this second taking notice of many particulars, relating to our design, we thought good to give it some place here; and to let the Reader know, that, besides an accurate Description of the two famous Cities of Indostan, Dehli and Agra, and many things discovering the Genius of the Mogols and Indians, as also those, which belong to their militia, &c. here is an account given, First, of the Extravagant opinions of the Gentiles of Indostan; of their odd belief concerning Eclipses; of the Books of Sciences received amongst them; of their Doctrine of the Transmigration of Souls, and of the Creation, Preservation and Destruction of the World; as also a Relation of the different Sects of Philosophers amongst them; of their Method of Physick, very different from ours, and how successful; of their ignorance in, and aversion from Anatomy; of their pleasant Tenets in Astronomy, Geography, and Chronology; of their opinion concerning Plants and Animals, importing that the Seeds of both of these kinds are not form'd anew, but were contrived in the First production of the World; as also, that they are actually the very entire Plant or Animal. Secondly, here is to be found a good description of the kingdom of Kachemire, the reputed Paradise of the East-Indies; its antient State; its present condition and excellencies for Soil, Vegetables, Waters, Cattle, Wild Deer; the wit and Industry of the Inhabitants in making fine stuffs, good Vernish, &c. the condition of its Mountains, one side of them being intolerably hot, and yielding Indian Plants, the other very tolerable, and affording fording none but European Plants; some remarkable about the Generation and Corruption of Trees there; strange Cascata's of water; a rare fountain flowing, and stopping regularly thrice a day, at a certain time of the year, and at other times flowing irregularly; and a great Lake having Ice in Summer, &c. The Situation of the kingdoms of Tibet, and the Commodities they offord. A considerable account of Voyages made by Caravans from Kachemire through the mountains of the great Tibet into Tartary and Cataja; and another, of the kingdom of Kacheguer, with directions how to travel thither from Kachemire, as also, how to pass from Kacheguer to Cataja. To all which is added an Answer to some Questions touching the Monsoons and Rains in India; the wonderful regularity of the Current of the Sea, and of the Winds there; the fertility of Bengale; and the Causes of the Inundation of the Nile and some other Rivers. The whole is concluded with a Letter concerning the Doctrine of Atoms and the nature of the Mind of Man. Historia & Meteorologia INCENDII ÆTNÆI, Anni 1669. Joh. Alph. Borelli. Regio Julio 1670. in 4o. Though we have seen several Relations communicated to us of this late fire Fire of Mount Ætna, one of which hath been inserted in Numb. 51. of those Tracts; yet will it be easily allow'd by those, that shall have read and considered this Discourse, composed and published by the famous Philosopher and Mathematician J. Alph. Borelli, that all others, that have hitherto written anything about it, must needs give the preeminence to him in respect of method, fulness, and philosophical reflexions; to the performance of which he averreth to have been induced by his Eminence the Cardinal de Medicis and the English R. Society. The Author then having given a short Topography of this Mountain, delivers first a succinct Relation both of the old and later Eruptions, as also of this last Fire of Ætna, and withall assigneth the perpendicular Height of the same, shewing it not to exceed three Italian miles, and here taking notice of Kepler's assigning two such miles for the height of the Atmosphere, and thence concluding the top of Ætna to be considerably raised above that region of the common Air; confirming the same by a known Experience, whereby those that are on the said top at a clear break of day, may plainly see the whole Island of Sicily, and all the Towns thereof, as 'twere elevated and hanging in the Air, near the Eye, just as, upon the account of Refraction, stones lying at the bottom of a pond appear nigh the surface of the water. After this, in the History itself of this Eruption he describeth particularly: First, the beginning of it, which happen'd on the 8th of March 1669, accompanied with Earth-quakes, and a Rent of the ground of 12 miles long, and 5 or 6 foot broad, as also with a terrible noise, roaring and cracking, vast globes of smoak first rising into the Air, and abundance of fiery melted stones being ejected soon after, which first ran like a flood of fire, and overwhelmed in a short space of time 13 towns, besides a part of the City of Catania itself, and afterwards were by the Air hardened into vast heaps of black and pumice-like stones, there call'd Sciarra; wasting and spoiling abundance of Vines, Olives, and other plants. The casting out of the Ashes and Sand continued for three whole months without ceasing, and filled all the neighbouring country, and cover'd all the Trees thereof for 15 miles about; the smallest dust flying even over Sea into Calabria by a Southwind, and into the most Southern parts of Sicily by a Northwind. But then on the 25th of March, by a new Earth-quake the Top or Turret of Mount Aetna itself fell in, whereby was made an Opening or Cauldron of three miles in compass, and vast quantities of new matter cast out, and amongst it, abundance of fiery sand, falling down with a yet burning heat at 8 miles distance from the Cauldron; whereupon the same by particular view and observation was found widen'd to the circumference of 6 miles. Meanwhile all considering men there were amaz'd at the force throwing out to so great a height such huge stones, whereof one was measur'd to be 60 palmes (or about 40 foot) long, which was faln down a mile from the Cauldron with that violence, that it was struck 30 palmes into the ground. When this fiery Torrent assaulted Catania itself, and had already by its impetuosity forced from its place a whole hillock, planted with vines, belonging to the Jesuits, and carried them floating, floating together with the soil bearing them, till it so swelled as to cover and sink them all; there appear'd some gallant persons, who by their ingenuity and extraordinary diligence, with fit Instruments, and raising vast strong walls, diverted the course of the fiery stream from that City, but chiefly by boaring through the stony heaps, and thereby making passage for that current another way, and turning part of it into the Sea, wherein it made a promontory of a mile's compass before the Town. It ceased by the 11th of July of the same year it began: And in May of 1670, our Author himself could handle without hurt the inner parts of the Cauldron and the former Torrent, and saw not so much as any smoak remaining in the highest part of that opening: Where yet he observeth, that notwithstanding this entire ceasing in the said places, there were yet found in several parts of this newly ejected Sciarra hot and strong-smelling fumes arising on high, especially near the walls of the South-side of Catania, where wells had been digged for watering their Garden fields. This being the Breviat of the History of this Eruption, which the Author describeth with many other considerable circumstances; we proceed to acquaint the Reader, that after this Description he expatiateth into divers important speculations and remarks thereupon. And first he comments upon the Form, Consistence, Bulk, and Motion of the ejected matter; taking particular notice of the great abundance of Sal Armoniac, that was found in all the holes and vents of the ground, and in the clefts of stones; and observing, that in three months time the fiery flood ran out 12 miles in length, and, taking the medium, one mile and two thirds of a mile in breadth, and stopp'd at last by the ceasing of new matter, and the condensation of the liquid matter by the cooling Air. Next, he compareth together the measure of the freshly ejected matter with that of the Mountain, and finds it to equal near a fourteen thousand part of the whole Mount. Thirdly, he evinceth, that Mount Ætna hath no such vast deep cavities, as some imagine, within its bowels near the surface of the Sea, but that there it is filled up with solid and stony matter, the huge weight of the superincumbent hill rather compressing it, than suffering any considerable hollownesses to continue therein. Fourthly, Fourthly, having exploded the imperfections of Meteorology delivered by the Antients of this Mountain, he discourseth of Subterraneous Heat, and deduceth the Cause of it from some concrete oleaginous and fatty substances, as sulphur, bitumen, and oyl, easily reducible into flame; examining withall both what is the origin of such sulphureous and bituminous matter, and how that comes to take flame: Where he digretheth to shew, how Gunpowder is set on fire, after he hath described the composition, and considered the stupendious force of the same by its percussion. Fifthly, he declareth, how and why the Fire in this Mountain was kindled; which he conceiveth to have been most probably done by such a way, as Quick-lime is heated by the diffusion of Water; whereupon he sheweth, how Earth-quakes, Flames, and Conflagrations may have ensued. Sixthly, he examins the origin and production of the fluid matter that was vitrified upon its being thrown out of this Mountain; and is of opinion (which he asserts by reason and experiment) that it was not any ignited and melted Sulphur or Brimstone, or both together, nor any metallic bodies that were converted into those vast stony and black masses, which they call Sciarra's, but rather Earth and Sand, together with some Alcalisat Salts burn'd by the fervent heat of the Ætnian furnaces, and so turn'd into a vitreous fluor, and afterwards, upon their being cool'd by the Air, into hard substances: Explaining withall, how the Asperity and Opacity of these stones are consistent with this sentiment. Seventhly, he discourseth plausibly of the Extent, Form and Scite of the Ætnian Furnace, and having, upon the examination of divers circumstances, found it but small and narrow, he labours to shew, how, this notwithstanding, so vast a quantity of matter, as amounts, according to his estimate, to about 100,000,000 cubic paces, could be therein melted and thence cast up. Eightly, he describeth the burning down and falling in of the highest top of Ætna. Ninthly, he inquireth into the Generation of those Sands abovementioned, and withall giveth an account, why in the New opening of Ætna there were heard such terrible and perpetual Thundering noises, as also, by what cause and force those Sands were thrown out; shewing withall, that they were really Sands, not *Aesop*, and solving Objections alledged against it. *Tenthly* he concludeth, that since, by his calculation, the upper part of Mount *Etna* hath been just so much depressed, as the mass of sand and stones ejected amounts to, this mass was furnish't by the Mountain itself, and from the Earth and Sands thereof produced and vitrified: where he taketh pains to make it appear, how all that prodigious quantity of matter, thrown out both now and in former ages, and seeming far to exceed the bulk of this whole Mountain, could be furnish't by *Etna* itself, and yet the same not be quite levelled with the ground: The like of which he considers of Mount *Vesuvius*. *Eleventhly*, he taketh particular notice, that, whereas the Fires were soon extinguish't in the place of the very Pit of the Sulphur and Bitumen, yet they lasted long in other places, that were waterish and destitute of combustible fatness; endeavouring to render a reason of this phenomenon. *Twelfthly*, he inserts an Observation of great plenty of *Sal Armoniac*, cast out in this *Incendium*, which, he saith, was seen adhering both to the corners and aperities of the *Sciarra*, and to the sides of the rents made in those huge stones, which the ejected matter was at length hardened into. And of this Salt he affirms, that there had been sublimed, (for he makes it *fatalitious*) to great store, that many thousands of pounds might be gathered; adding, that even a whole year after the extinction of the Fire in the Mouth's of *Etna*, there were found remaining divers vents about *Catania*, exhaling store of smoak, which had the like *Sal Armoniac* sticking to the sides and edges of the Stones. *Lastly*, he rejecteth the opinion of those, that will maintain the *Aetnean* Fires to have been perpetual and never extinguish'd, asserting the frequent cessation of them, and withall assigning the cause of that cessation, as well as that of their renovation. Concerning which, and many other considerable remarks and reflections, too many to be here recited, we refer the Reader to the Book itself. *An Advertisement*