An Account of Some Books

An Account of some Books: I. ELEMENTS des MATHEMATIQUES, ou Principes Généraux de toutes les Sciences qui ont les Grandeurs pour Objet; par J.P. A Paris, 1675, in quarto. The Author of this Work makes it his business to deliver a short and easy Method to compare Quantities, and to discover their Proportions and Relations to one another by Characters of Numbers, and Letters of the Alphabet; affirming to have here demonstrated things in a Geometrical order, and rendered the Algebraical Analysis much easier, and treated the same more fundamentally than hath been done hitherto. By Quantity he understands here not only the Extension in length, breadth and depth, but whatsoever we conceive to be capable of more or less, and that can be exactly measured, whether it be exactly known, or supposed such. Thus Time, Weight, Celerity, and even the Sensible Qualities, the Degrees of Perfection, being capable of more or less, are, to him, the Object of the Mathematicks. For, faith he, if you do exactly know these perfections and qualities, you may compare them in order to know exactly their proportions, and if you do not know them exactly, you may compare them by supposition: For, if you know, that a piece of Iron (e.g.) is four times heavier than such a piece of Wood; by supposing that Wood is a thousand times weightier than Air, you may conclude by this supposition, that the Iron is 4000 times heavier than Air. He considers, that though Arithmetick be a Science from which all others depend; yet 'tis this Algebra, which serves to elucidate, extend and perfect, as much as is possible, Arithmetick, and generally all the Sciences that relate to the Mathematicks: It being so general, that it considers all Quantities, and what it demonstrates being capable to be applied not only to Numbers Lines and Figures, Weights and Velocities; but also to all such Numbers, Lines, Velocities, and particular quantities, as you can conceive in each species of Quantities. But 'tis not only the Extent and Universality of Algebra, for which he commends it; but also the Facility it affords to the mind of discovering the most hidden Verities, and which 'tis impossible to manifest by ordinary Arithmetick and Geometry, or by the aid of any other Science; since it not being possible to give to our understanding a greater extent and capacity than it hath, this Science only teaches to manage it, by representing to it under very short expressions an Aggregat of many Ideas, by taking it up so little by the senses. senses that it leaves it in a manner altogether to itself, and by assi- ting it to run through all the proportions of Quantities which it examines, in a dextrous, expedit and easy manner: So that nothing escapes the understanding in the subject under consideration; and the clear and distinct neatness of its ratiocinations always disco- vers to it the shortest way of researched Truths, as many of them as it can come to know, or the means which it wants to attain them, if it cannot know them. He takes notice, that, there being particular Sciences that de- pend from Geometry, there are those that consider the same as the general Principle of all knowledge: And that, forasmuch as Geo- metry is pleasing enough by reason of the Figures that fall under the Imagination, there are many that do inconsiderately prefer it to Algebra; and that they imagine withal, that the Geometrical De- monstrations by Lines are the only true ones, because they make themselves as 'twere sensible. To this he faith, that he is not igno- rant, there are things peculiar to Geometry that must be known and demonstrated by Figures; but that, to handle this Science as it ought, we are often obliged to make use of Algebra, and that, be- cause the proofs thereof are the most general and the most simple, they are therefore to be accounted the most Natural Demonstrati- ons. And if it be objected, that Incommensurable Quantities cannot be discovered nor expressed by Numbers, but they always may by Lines, and so Geometry is more exact and of a greater extent than the Science of Numbers: He answers, 1. That Incommensu- rable Quantities may always be expressed by Incommensurable Numbers; and if the Incommensurable Numbers are not altogeth- er known, 'tis because the Incommensurable Quantities, imply- ing somewhat of infinite and incomprehensible, are not capable of being fully known. 2. That Lines are never the true expressions of Incommensurable Quantities, nor even of the Commensurable, forasmuch as that which maketh the quantity not known, cannot be an expression thereof; and that the lines, of which the Geome- tricians pretend to express the unknown quantities, do not make known their quantities. He grants it to be true, that Geometri- cians do demonstrate, that those Lines are equal to those Quan- tities; but he adds, that those lines themselves are unknown to the Understanding, though they are known by the Eyes or by the Im- agination; and that, if you would have expressions speaking to the Mind and not to the Eyes, you must recur to Incommensurable numbers: numbers: So that these numbers are still more known than those lines, seeing they do better express and represent them to the Understanding. As (e.g.) this number $\sqrt{20}$ is much more known, than the Hypotenusa of a right angle of which the sides are 2 and 4; for, 'tis at least known, that $\sqrt{20}$ is about $4\frac{1}{2}$; and if you will know it more precisely, you may, by the rules of the Approximation of Roots. But you know not the bigness of the Line that sustains a right angle, though you see or phancy it. He observes further, that the Analytical (which is the principal) part of Algebra, is incomparably more fertile for the discovery of Truths, than Figures, and that without it 'tis in a manner impossible to resolve an infinity of Problems. For, faith he, how can any man imagine that long concatenation of embarrassing Lines and Figures, where you ought to see distinctly so many different proportions and respects, before you know what it is that the resolution sought for do immediately depend upon. Now, as to the Order which our Author hath observed in those Elements, they are divided into two Parts. The first, containing five books, explains and demonstrates both the Supputation with Numbers, which is otherwise call'd Arithmetick, and that of Symbols or Letters, which is called Algebra. The other, in four books more, expatiates and treats fundamentally of the Analytical part, teaching to resolve Questions, and to discover the General Truths of the Mathematicks; that is to say, those which regard Quantities generally taken, yet without supposing other knowledge than what is granted; but making use of those Operations only that are establish't in the first part. In the first book of all, the Author shews, that an Unit and Numbers are the sole Idea's, by which we can regulate the measure of Quantities, and exactly determine what is knowable of them. And after he hath explained the fundamental Idea's that serve us to compare quantities amongst themselves; he teaches in the sequel of this book the four first Operations that are made by Numbers or Entire quantities, which are considered as Proportions, whereof the first term only is expressed, and the second, which is always an Unit, understood. The second book is of the same Operations upon Fractions, which are Proportions of quantities, of which every term is expressed. The third is of Powers and their Resolutions, whereof all the Rules are included in one only Problem, by means of a Table that represents in an Epitome all those Rules with their Demonstration, tion, after a manner that is not less general than simple and easy to understand. And since this Resolution of the Powers doth not always afford Commensurable quantities, or such as are exactly known, but sometimes Incommensurable ones; these are explicated in the fourth book, together with all the Operations that are made concerning them. The fifth treats of the Comparison of Proportions: Which part he finds to be so vast and fruitful, and the uses thereof to be of such an extent in most Sciences, that there are few, if any, that can be well taught without it. The Geometrical Equalities and Proportions, which are one sort of the genus of Equalities, are the things that render this part so considerable, and for the elucidating of which our Author hath most of all employed himself in this work; and the four last books of the second part are nothing but a sequel of what hath been said of the Equalities in the fifth book of the first part. Now in the said four books he settles first the Grounds of Analysis. Next, after having there given some idea of the method of Diaphantus, and of that of Vieta, he is particular in explaining the method of Des-Cartes, which he esteems to be the most general, the most fertile, and the most easy of all. Yet seeing that this famous Man hath not demonstrated, nor so much as explained, all the Principles which he hath made use of, our Author intimates, that the Reader will not find in his Writings the same advantages for understanding his Analysis, as may be had from these Elements. For, after he hath clearly explained and demonstrated all those Principles, he thence deduces in order not only all the Discoveries made by Des-Cartes, but also other new and more useful ones. For it may be seen in the last book, that those new Discoveries furnish Rules that are much shorter than the Cartesian, and one may even draw analytically from them much of certain and very universal knowledge, which he did not believe could be discovered without the aid of Parabolical lines, or such other as belong to the Geometria composita, as the Hyperbolick, &c. But, forasmuch as the Author esteems that these Elements are principally written for Beginners, and even such as have not so much as the knowledge of Arithmetick, he desires that such Readers would have their pen at hand, to make themselves the operations of all the different Examples, deliver'd in great number, his aim aim being to accustom them to practise the Rules, and to make those things familiar and sensible to them, which at first seem to be abstract and difficult enough, especially to those that are not yet accustomed to the exercise of their Understanding. As for those that are already versed in common Arithmetick and Algebra, they, he faith, have discerning enough to exempt themselves from reading what they already know. Though he hopes withal, that there will be those that will not find it tedious to peruse all, that so they may observe the connexions, which possibly they had not yet observed, between all those Truths and the different parts of the Mathematics; and to establish also their knowledge upon principles, that may perhaps appear to them more plain and more natural, and in less number than those which they have used hitherto. II. De l'ART de PARLER; à Paris, 1675. in duodecimo. As there was printed at Paris, some years since, the Art of Thinking, (whereof an Account was given in Numb. 106. of these Tracts,) so there is lately published in the same place this Art of Speaking, of which some Description is intended here. This Author doth not, as is ordinary, crowd his book with a heap of Precepts, whereby the Memory is burden'd and the Mind embarrassed; but, like a good philosopher, makes it his chief business to teach the Ground of the Art he treats of, and its Natural Principles, which being well known, there needs not that multitude of Rules, which slip out of the Memory almost as soon as they are entered. Now to make the Reader comprehend the true Reasons of the Principles of Rhetorick, he begins with explaining, How Discourse comes to be formed; and there being nothing better than Nature herself to teach us the form that our Words ought to have for expressing our Thoughts and the Motions of our Will, he represents to himself a Troup of Men newly born and that never have spoken before. He considers what these Men would do: He shews, that being soon tired with expressing their mind to one another by Gestures, they would quickly find the advantage of Speaking, and form a Language to themselves: He inquires, what form they would give to that language; and in this inquiry he lays the Foundations of all languages, and renders the Reason of all the Rules prescribed by Grammarians; shewing that this research is very very useful to learn Languages with more ease, and to speak them with more exactness. And having made these New Men at their part, he declares, what hath been the true Origin of Tongues, and that 'tis not Hazard that hath made men find out the Use of Speech; yet shewing withal, that Speech is subject to mens Will, and that Custome or the common Consent of men exercises an absolute power over Words: Whereupon he gives Rules to know which are the Laws of Custome, and to observe them, after he hath instructed his Reader which are the Laws prescribed by Reason. And these are the Contents of the first of the Four parts of this Book. In the second part he observes, that the most fertile Languages are not able to furnish terms proper to express all our Ideas, and that therefore we must have recourse to Art, borrowing terms from things that are in a manner alike, or have some connexion or relation with those that we would express, and for which the ordinary use doth not afford us proper words: Which borrowed Expressions are called Tropes; of all the kinds and uses of which he treats at large. In the same part he takes notice, that as Nature hath so dispos'd our Body as that it falls into postures proper to shun what may hurt it, or into those that are fittest to receive what may do it good; so the same leads us to take certain turns in speaking, capable to produce in the minds of our hearers the effects we desire, whether it be Meekness or Choler, Hatred or Love: Which terms are called Figures; of which the Author treats with a particular diligence, not being content to give us their Names with some Examples, (as is vulgarly done) but also teaching us the Nature of each Figure, and the Use to be made thereof. Next he considers in the third part, that forasmuch as the facility with which men speak, and the pleasure that a discourse well pronounced begets, have brought men rather to make use of Speech to signify their thoughts, than of any other Sign; they have therefore studied to find in the ranging of Words, that which makes a discourse to be utter'd more easily, and to be heard more delightfully. And hereupon the Author enlarges himself by shewing, what is to be avoided, and what is to be observed in the disposition of the words to facilitate the Utterance and to please the Ear. And here it is that he treats of the composition of Periods and of the Art of Versification; and having shewed what it is that can please the Ear in the sound of words, he adds, how the Rules prescribed by Masters for composing Periods and making Verses in all languages, ges, have made for their end the rendering the delivery easie and pleasant. In the fourth part he treats of Styles or Manners of speaking, diversified according to the several inclinations and natural dispositions of Men. Here he proposes his advice for regulating a Style; and, because every matter must be handled in a way suitable to it, he teaches how to raise or depress, sweeten or asperate a style according as the nature of the Subject requireth. On which occasion he examines the quality of the Style of Orators, Poets, Philosophers, Historians. He concludes the Book with giving us a very fair Idea of the Art of Perswading, which is very different from the Art of Speaking, in regard that not all those that speak well know the secret of gaining hearts. In the whole, and particularly in the discovery made of the nature of the Art of Rhetorick, there are to be found considerable reflexions upon our Mind (of which Speech is an Image,) which may contribute not a little to bring us to the knowledge of ourselves. And every Curious Spirit will doubtless be pleased to learn to know the reasons here given of all the Rules, which the Art of Speaking prescribes; This Author, when he treats of what it is that pleaseth in a discourse, not contenting himself with saying 'tis something I know not what, but naming the very thing, and leading the Reader to the very source of that pleasure, and making him understand the very principles of those Rules, which those do follow that give delight in Speaking, &c. III. The manner of raising, ordering and improving Forrest Trees: Also how to plant, make and keep Woods, Walks, Avenues, Lawns, Hedges, &c. with several Figures proper for Avenues and Walks to end in; and convenient Figures for Lawns: Also Rules and Tables, shewing How the ingenious Planter may measure superficial Figures; With Rules, How to divide Woods and Land; and how to measure Timber and other Solid bodies, either by Arithmetick or Geometry, &c. By M. Cooke; in quarto. I. In former Times, not only Princely and Noble Palaces, and Seats of Honour, but also generally the Mansions of the Gentry, all over England, were adorned with Groves on the next Hills; Hills; and, in nearer approaches, with goodly rows of sturdy Oaks, tall Elms, huge Chesnuts, and other stately Trees of English production: To defend the Avenues, Gardens, Orchards, Walks and Ridings from violent Winds and Storms in the roughest seasons; and for cooling refreshments in the Heats of Summer: And this was a credit and shelter in all fair Villa's, which are said to be more in England, than on so much ground of any other part of Europe. And it was believed, that these long-liv'd Vegetables had some affinity, congruity, congeniality or propension to sustain, cherish and lengthen the lives of Mortals; and to have somewhat of the nature of Perennial Fountains; to retain (round about them) a degree of Warmth in Winter Frosts, and of coolness (beside their shades) in Summer Heats. Of Gardens and Orchards Mr. J.W. faith, p.147,148. The exercise of Planting, Graffing, Pruning and Walking in them, very much tendeth to salubrity, and to cure several distempers incident to our Natures (as hath been experienced,) and towards the prolongation of life. And if any of these Vegetables participate of the nature of smoaking Fountains, it may reasonably be apprehended, that they may attemper and mitigate the extremities of both Seasons, to be more agreeable to Humane Constitutions. 2. And 'tis a real Truth (though I must here be more sober than to mention it in good earnest,) that (sometimes) the stateliest Trees will familiarly treat, and answer distinctly to all the Discourses, Noise and Voices of the Family, from the softer whisper to the loudest railery, with vocal imitation. And they are so perfectly Musical, that they will keep Time and true Consort to any Tune, from the highest Treble to the deepest Basse: And 'tis a great pleasure to the Musical (which are alwaies the best) Natures, to enjoy a Musical Habitation. 3. And seriously this Vegetable Furniture was held a sure Mark, to distinguish, at great distance, Good Husbands and Hospitable Householders, from Wasteful and broken Consumers of their Patrimony. 4. And much is done lately, both for Ornament and Healthfulness, about the Amenities of the Royal Palaces, and of many other chief Mansions, by planting the most beautiful, wholesome and verdant Trees, in all their Avenues, Walks and Ridings. 5. And much more may yet be done for the Credit and Reputation, as well as for the healthfulness and pleasantness of England, if we proceed to follow the best examples, for places of Ancient Honour; Honour, for Cities and Towns; and especially for both our Famous Universities, where they have many Colleges with spacious Walks and fair Gardens. 6. And for our choice of the most convenient Plants, ingenious Muffet (in his third Chapter of Healths Improvement) learnedly instructs us, by the experience of many Ages and Nations, to avoid some Plants of a noxious and poisonous breath, and to adopt Health-breathing Plants; of which benign kind great variety is now found by many Trials to agree with our Soil and Climate. So that our Universities have no necessity to yield to the Reputation of Leyden for their Aquatic Arbors, since ours may soon be furnished with more wholesome, beautiful and fragrant Blossoms and Evergreens. And more Essays grow on apace. For all these Concernments, I could do no less than give publick notice of the seasonable Aids and Encouragements herein afforded by this our skilful and industrious Author. Postscript. If among their Adorning-Trees due care be had for the planting of Mulberry-trees, in the approaches of Cities and Towns, they may do well in time for many Poor in England; as the care for the same hath brought a vast Treasure into France. IV and V. The French Gardiner reprinted; to which is annexed the English Vineyard vindicated, and the Way of making and ordering Wines in France, &c. For another Concernment I must again give notice, that The French Gardiner, which gives proper Instructions for the culture and propagation of the best Esculent Plants, (which are yet much wanting in England,) is reprinted in a third Edition, Illustrated with Sculptures: To which is annexed, The English Vineyard vindicated; and, The Way of making and ordering Wines in France: in octavo. Sold by Mr. Tooke at the Ship in Paul's Churchyard. "The Fruits of the Earth, and especially of Trees (faith Mr. J.W.p.148.) were the first food ordained for Man to eat; by eating of which (before Flesh became his meat) he lived to a greater age than any since have been observed to have lived." Errata in Numb.125. P.602.l.32. for your read the, ib.l.36.r. singeing. London, Printed for John Martyn, Printer to the Royal Society, at the Bell in St. Pauls Church-yard. 1676.