Accounts of Books

Accounts of Books. (1.) Le grand & fameux Probleme de la Quadrature du Cercle resolu Geometriquement par le Cercle & la Ligne droite, par Monsieur Mallement de Messange. A Paris, in 12° 1686. With a Refutation of the same, by Mr. D. Cluverius. Reg. S. Soc. This Author is one of those unhappy Geometricians, who without having acquired a through Understanding of the Principles, have yet thought themselves able to master the abstrusest Difficulties in this nice Mathematical Science, where the least oversight or mistake subverts the whole superstructure. Hence it is, that the true Quadrature of the Circle here pretended to, is lost upon the same Rock with those many others, which the less knowing and more opinionated of their own Skill have produced, in this and the last Century: But briefly to shew wherein the Paralogism of our Author consists, we must first lay down the construction, whereby he pretends to do the Business: In Tab. 2. Fig. 2. let $f k \approx k$ be a Circle, $f a \approx$ the Diameter, $a$ the Center, $k \approx k$ an equilateral Triangle inscribed, $B b$ a line equal to the three sides of that Triangle, and dividing the Arch $f k$ equally in $i$, the line $i e$ will be half the side of a Hexagon inscribed, which side taken 6 times, is the line $e E = \text{to the circumference of the Hexagon;}$ and dividing the Arch $i f$ in $b$, the fine $b d$ is half the side of the Dodecagon inscribed, and $D d = 24 b d$ is the circumference of the Dodecagon; and proceeding after the same manner, the circumferences of Polygons of 24, 48, 96 sides, &c. may be found, approaching still nearer and nearer to the circumference of the Circle, which at length will be equal to the line $f F$ in the Tangent; but how to find the Point $F$ is all the Skill: Here our Author tells us, that the Points $BEDF$ are all in the Arch of a Circle, whose center is in the line $H h$. f z continued; but to contract his Work into a little compass, he assures us that it is all one, if instead of the whole circumferences B b, E e, D d, F f he take the lines b q, e p, d o, f m, each a third part of its correspondent, and that in this case too, the Circle whose Center is L, taken in the Diameter f z continued, shall pass through the Points o, p, q and intersect the Tangent F f in the Point m, so as to leave f m = to a third of the circumference of the Circle; which supposition being proved to be groundless and erroneous, all the consequences drawn from thence must be so too. If our Author had but considered what the intermediate Points of the Curve between o & p, p & q, q & t ought to express, he could not but have discovered the fallacy himself, for the lines o d, p e, q b are each proportioned to f m, as the fines k b, i e, h d to their respective Arches k f, i f, h f, and so of all the rest between o and m. This would have taught him, that the Curve he has occasion to use, did universally express the Proportion of the Arches to their Sines, by that of the line f m to its respective ordinates; that it was a sort of Linea Quadratrix, to be reckoned among the Lineae Geometrice Irrationales, or such whose relation between the parts of the Diameter and the Ordinates, are not generally expressible by any one Equation; that this Curve did intersect the Circle in the middle of the Arch k z, and continue it in the point z. This Curve will be better understood in Fig. 3, where it is drawn as it ought, and wherein the proportion of the line f m to the lines q b, is as any Arch k f is to its Sine k b. Tis evident, that this Curve is not the Circle m o p q t in Fig. 2, yet tis not apparent but that a Circle passing through the point m, may intersect it in several points, as o, p, q: (but to suppose it to pass through all the extremities of the Circumferences of the infinite Polygons between the Circle and Triangle, or their thirds, is to make it coincident with that Curve.) It remains therefore to shew, that the Circle passing through p and q, whose center is in the Line f z, does not pass pass through the point \( o \), which from the following Considerations will be made evident. First let it be required, by the extremities of the lines \( a, b, c \), or \( b, g, f \) (in Fig. 4) parallel one to another and \( d, e \), or \( k, l \), given as parts or segments of the Axis or Diameter of the Figure, to determine what curvity passeth through their extremities, according to the conditions of the five Conical Sections. First if it be found that \[ \frac{b^2 - aa}{de} + \frac{cc + aa}{de + ee} \] is equal to 1 = \[ \frac{cc - aa}{dd + ed} + \frac{cc + bb}{ed} \] then it is the Characteristic of a Circle, the Lines \( a, b, c \) being disposed in an uniform increasing order: But if \( c \) the biggest stands in the middle, then \[ \frac{cc - aa}{dd + ed} + \frac{cc - bb}{ee + de} \] will shew the same. If the Lines \( d, e \) be segments of a line drawn parallel to the Axis, then transposing and ordering the foregoing Equations, Rules also may be found accordingly. If \[ \frac{c - b}{b - a} = \frac{e}{d}, \quad \text{or} \quad \frac{c - a}{b - a} = \frac{e}{d} \] = 1, then the Line passing through the Extremities is right: If \[ \frac{cc - bb}{bb - aa} = \frac{e}{a} \] then a Parabola is designed. The Characteristic of an Hyperbola or Ellipsis differs not from that of a Circle, but only by a Relation to the inequality of the Axes, and the alteration of the Signs \( + \) and \( - \). Secondly, out of \( a, b, d \), or \( b, c, e \), lines given, that stand in the Arch of a Circle, to find the Distance from the Center = to \( m \), or \( m + e \), and to determine the Radius. There is a little variety in the case, when the given lines are in the same Quadrant or otherwise: but there being only occasion for this first Case, the Rule is this, \[ \frac{cc - bb}{2e} + \frac{e}{e} = e + m: \quad \text{And} \] \[ \sqrt{\frac{cccc - 2ccbb + bbbb}{4ee}} + \frac{e}{e} + \frac{bb}{e} + \frac{cc}{e} = \text{Radius}. \] Thirdly, in a Circle, having \( a, b, d \) and \( e \), to find \( c \): The Equation is \(\frac{b^2 e - a^2 e}{d} = de - e^2 + b^2 = c^2\). Fourthly: To inscribe Polygons in a continual double Progression within a Circle, many different Rules may be given: the following will serve, which is the same with that, how to find the subtense of an Arch, out of the subtense of a double Arch. The Rule is thus; \(2 R^2 - \sqrt{4 R^4 - A^2 R^2} = B^2\); Supposing \(A\) to be the Chord of a double Arch, and \(B\) of a single Arch. From hence it is easily deduced, that \(\sqrt{3} R^2\) being the side of an equilateral Triangle inscribed, the side of a Hexagon will be \(R\): of the Dodecagon \(\sqrt{\text{un}} \frac{1}{2} R^2 - \sqrt{\frac{1}{4} R^2}\): and so for the rest. Now reducing according to these Equations the Lines to Numbers, it will be found that in Fig. 2 Tab. 2 \[ \begin{align*} bq &= 173205.08 \\ ep &= 200000.00 \\ do &= 207055.23 \\ ab &= 50000.00 \\ eb &= 36602.54 \\ de &= 9990.04 \\ fm &= 209439.51 \end{align*} \] But supposing, as our Authour will have it, that \(do\) stands in the same Circle with \(bq\) and \(ep\), it follows that the square of \(do = 422638679\) &c. whereas it should have been equall to \(428718707\) &c. Square of \(do\) in the Table. The Square of the Tangent \(fm\) is also a great deal too small, and the whole Quadrature too little: All which make it appear, that the Glory of Lewis the Great is not (as this Book pretends) much advanced by the Achievements of this Author; who would have done well, in a Matter that so little needed it, to have forborne to make use of the sacred Words of our Saviour, Math. 11th. 25th. II. Voiage de Siam des Peres Jesuites envoyez par el Roy aux Indes & a la Chine. A Paris 1686. 4°. This is a second Relation of the Voiage and Embassy of the French to the King of Siam in the Year 1685, and being a more particular Account than the former, an extract of this, 'twas thought, might suffice for both: That was composed by le Chevalier Chaumont the Embassadour, and now this by le Pere Tachart Jesuite, who was one of six Fathers of his Order, that went with the Embassadour, as Missionaries to China. The whole being much interspersed with matters of Religion and Ceremony, I shall only take notice of such things as relate to Arts and Sciences, and particularly of the Astronomical Observations made at the Cape of Good Hope and at Siam; whereby the Longitudes of those places are stated: following herein the Authors method. He divides his Treatise into six Books, whereof the first contains the Voiage from Brest to the Cape of Good-Hope. Here he gives the reasons and motives of sending this Embassy, as likewise the six Jesuites who are Mathematicians and by the Kings Letters Patents are so stiled; Their Instructions being, besides their Spiritual Function, to prosecute the business of the Royal Academie of Paris (of which they are admitted Members) by accurately observing the curious things in Art and Nature, and particularly to make Observations for discovering the Longitudes of the Places where they pass; for which purpose they are well provided with Instruments. They sailed from Brest on the third of March ft. n. and arrived at the Cape of Good-Hope the last day of May, taking notice by the way of the several remarkables in that Voiage, which are here too well known to need repeating; But mentioning the faultiness and rectification of the Southern Constellations, our Author is not willing to take notice of what has been done in that matter by a member of of the Royal Society of London, tho' his Catalogue of those Stars hath been translated from Latine into French and Printed at Paris, and an Account thereof is in the Journal des Scavans of Aug. 7. & Sep. 4. 1679; but speaks of it as a thing not done, wishing they had had the opportunity to augment the Science of Astronomy, by observing them themselves. The second Book is entituled the Voyage from the Cape of Good-Hope to the Island Java; but is chiefly taken up with the description of the Colonie of the Hollanders there, the Natives, and the Astronomical Observations they made there during their stay, by which they have determined the Longitude of the Cape of Good-Hope 18 degrees to the East of Paris: (but we here must begg leave to make a Remark) He mentions several Nations of the Natives, viz. the Hottentots, whom he describes at large, the Namaquaas, (of these two there are the Figures; the Ubiquaas, the Gouriquaas, Ilafiquaas, Sousiquaas and the Odiquaas: and here he relates a Voiage made in the Year 1685, as farr as the Tropick, by the Governour of the Cape, Mr. Vanderstell; who is said to have found about the Latitude of 27 degrees and about 10 or 12 leagues from the shore, a Nation of Natives that are very Musical, who have long Hair flowing on their shoulders, some of the Men as White as Europeans, and their Women Naturally very White, but they Blacken themselves to please their Husbands: This Nation seems to have much more Intelligence than their Neighbours, but some circumstances seem to argue it Fabulous. Here are likewise the Figures of a Stagg with Horns like a Goat, of the Zembra, the Sea Cow, the Cerafes or Horned Serpent, the Camelion, and two sorts of uncommon Lizards, whereof one is made to have 3 fair Crosses on his back: Speaking of Elephants he says he was told by creditable persons, that they had seen the foot steps of Elephants two feet and half diameter; and that there are Rhinocerotes there as bigg as ordinary Elephants, but, by I know not what mistake, he makes the Rhinocerote a two horned Animal. The Voi- age from the Cape to Java (in fight whereof they arrived Aug. 5th.) contains nothing very Extraordinary. The third Book is the Voiage from Java to the Kingdome of Siam: which is chiefly taken up with what occurred at Bantam and Batavia, and at their arrival on the Coast of Siam, here are represented the Roads of Bantam and Batavia, together with the Plan of the City and Fortress of Batavia. The fourth Book describes the Entry and Audience of the French Embassadour at the Court of Siam, who, as they say, was received with more Honour and Respect, than was ever yet shewn to any Ambassadour whatever; and that even those of Persia, the Mogul and the Tartar Emperour of China, (tho' his neighbour, and by much the most Potent Monarch of the Universe) present themselves before the King of Siam on their Knees, whereas Mr. Chaumont the French Ambassadour made his harangue, sitting with his Hat on his Head. Here are described the Balloons or Barges of State which are used at Siam, which are of a very odd Figure, as of Serpents or Sea-Horses, but which by their Sharpness and number of Oars are of an incredible Swiftness: here likewise 'tis related that the old White Elephant of the King of Siam is near upon 300 Years Old; as also that there are Tumblers there, of an extraordinary Agility, as that they would stand upon one Foot on the top of a Bamboo of 80 or 100 Foot high, and then turn themselves, and stand on their Heads thereon, and afterwards hang, by the Chin only, on the top of the same, and then descend by a Ladder down right, with an incredible Swiftness, working their Bodies all the while through the Rounds of the Ladder. The fifth Book is entituled the Return of the Voiage of Siam, and first relates several notable Shews presented to entertain the Ambassadour; as the fight of two Elephants, who were only suffered to twist each others Teeth, as Bulls do their Horns; the fight of an Elephant and a Tiger or rather a Panther according to the description; and the manner of catching the wild wild Elephants, by alluring them into an inclosure by the means of a Female tame Elephant, and the like. Next are related several Observations of the variation of the Magnetical Needle, which was found towards the end of the Year 1685, to be about half a degree West, at Louvo near Siam; as likewise the Observation of an Eclipse of the Moon on the 10th December st. n. post medium noctem, made at a place near Louvo called Thlee-Poussonne, in the presence of the King. It begun about 15h. 20m. the total Darkness at 16h. 23m. 45s. the Emerion or end of Total Darkness at 18h. 2m. 36s. or rather, as is there said, at 18h. 10m. 25s. whereby the Longitude of this place is found 93 degrees and half from Paris, and about 6h. 45m. to the East of London, as may be seen by comparing this Observation with the Observations thereof made at Dantzick, Nurenburg and Lisbon, Published in Philos. Transact. Num. 178, 182 and 184. And whereas tis here said that some Charts have made the Longitude of Siam above 20 degrees more than it is, tis to be understood only of the Charts of Sanfson, which in this particular are the worst extant: But that this Correction is just, we are fully satisfied, by the like errors in those parts, discovered and published in the Philosoph. Collect. of Feb. 1684, and in Philos. Trans. of June 1683. The relation of the homeward bound Voyage (which was of about 6 Months) is short and contains very few considerable Remarks. The sixth and last Book is of the manners and Religion of the Siamesers, where is a short description of the Kingdome and Capital City of Siam: Next the habits of the People, and their use of Betel, Arek, and Tea is described, as likewise the Root Ginseng so much esteemed in the East, with its Vertues and Uses. As to the Religion of this people, which is here described at large, I shall say little to it, as not falling under our Argument, only one principal point therein is the Metempsychois of Pythagoras and the Bramines, and they hold the Eternity of the World, but on the contrary they suppose God Mortal, Corporeal and produced in time: their present God they call Sommonokbodom. A Remark concerning the Longitude of the Cape of Good Hope. In the second Book of this Voiage are related two Observations of the Satellites of Jupiter, capable, if well made, to ascertain the Longitude of the Cape of Good Hope. The first was there made June 2d. ft. n. 1685, when at 11h. 29m. 20s. the first or innermost Satellite touched the Western edge of Jupiter and at 11h. 30m. 50s. it appeared no more: this Observation is said to be made with an excellent Telescope of twelve Foot: The other was on June 4th. following ft. n. when the Emerion of the same Satellite was observed at 9h. 37m. 40s. from which latter is concluded, that the Longitude of the Cape is 18 gr. to the East of Paris, for that the said Emerion, according to the Calculus of Sigr. Cassini, in the Meridian of Paris ought to have happened at 8h. 26m. This same Emerion is computed by Mr. Flamsteed at 8h. 19m. at London, that is, 3m. later than by Sigr. Cassini; and considering that neither is verified by Observation in Europe, the Longitude hence deduced is doubtful at least 3 minutes, if this had been the only Observation: But the former being considered will yet shew that there is a much greater doubt still remaining: For from certain Astronomical principles the parallax of the Orb, or difference between the place of Jupiter seen from the Sun and Earth was, at the time of the first Observation, 9 gr. 19m. which Arch that Satellite moves in 1h. 6m. and the utmost duration of an Eclipse thereof in this position of Jupiter being scarce 2h. 20m. (as appears by the accurate Observations of Mr. Cassini and Mr. Flamsteed) it will follow, that from the Immersion behind Jupiters Western Edg, to the Emerion out of the shadow, there could not be full 3h. 26m. wherefore the Emerion out of the shadow, on June 2d. ought, according to the time of the Immersion, to be at 14h. 56m. at latest, at the Cape; which by Mr. Flamsteeds Calculus was at London 13h. 51m. or according to Sigr. Cassini at 13h. 58m. at Paris. Hence the Longitude of the Cape will be found but 14 degrees and half at most to the East of Paris; so that these 2 Observations will differ in the result about a quarter of an hour; which is a little too much. However there are some reasons that seem to argue for this latter Longitude rather than the former; for it is much easier to observe what becomes of a luminous Object that appears, than to wait upon the first appearance of a Star Eclipsed: and tis probable that the Satellite might, in the latter time, be several minutes Emerged out of the Shadow, when they might first perceive it; but they could not but see the application to the Body of Jupiter in the former, if we may suppose their Telescopes so good as they are said to be: And that the Cape of Good Hope is not more than an hour to the East of Paris, is proved by the constant consent of our Navigators, who find by their Reckonings that the Island of St. Helena is about 22 or 23 degrees of Longitude to the Westward of the Cape: (and that Sailing both backward and forwards tis the same, which takes away the Objection of Currents) now by accurate Observations made at St. Helena, and compared with others made in Europe at the same time, the Longitude of that Isle is certainly about $8\frac{1}{2}$ degrees to the West of Paris: It follows therefore that the Cape cannot be much more than 14 or 15 degrees to the East of Paris; and undoubtedly it must be less than 18, for 3 degrees is much too great an Error to be committed in so short a distance Sailing.