Thursday Decem. 10. 1674. At a Meeting of the Council of the Royal Society. WHereas it was desired by the Royal Society, that a Discourse made before them by Sir William Petty Knight, at their Meeting the 26. of November last, might be Printed: It is this day Ordered by the Council of the said Society, That the said Discourse be Printed by the Printer of the Royal Society. BROUNCKEK, P. R. S. THE DISCOURSE Made before the Royal Society The 26. of November 1674. Concerning the Use of Duplicate Proportion In sundry Important Particulars: Together with a New Hypothesis of Springing or Elastique Motions. BY Sir WILLIAM PETTY, Kt. Fellow of the said Society. Pondere, Mensurâ, & Numero Deus omnia fecit: Mensuram & Pondus Numeres, Numero omnia fecit. LONDON: Printed for john Martin, Printer to the Royal Society, at the Bell in St. Paul's Churchyard, 1674. To his Grace, WILLIAM, LORD DUKE OF NEWCASTLE. May it please your Grace, I Am commanded by the Royal Society to Print the Discourse, which I made before them, upon the last Meeting-day of their last year, and next before that of their Anniversary Election: Because, as Drapers cut Patterns of their whole Cloth out of an End, not because the End is better than the rest, but because it may be best spared; so (I suppose) the Society are content, that this Exercise pass for a Sample, pro tanto, of what they are doing; for that the same may be conceived to consist of three parts, viz. The first being an Endeavour to explain the Intricate Notions, or Philosophia Prima of Place, Time, Motion, Elasticity, etc. in a way which the meanest Member of adult Mankind is capable of understanding: The second being, to excite the World to the study of a little Mathematics, by showing the use of Duplicate Proportions in some of the most weighty of Humane affairs, which Notion a Child of 12 years old may learn in an hour: And the last being, without Chimerical Speculations, to consider such points and properties, even in Atoms (such, whereof perhaps a Million do not make up one visible Corpusculum,) as may give an intelligible Account of the Nexures, Mixtures, and Mobilities of all the parts of the Universe. In like manner, 'tis the Profession of the Society, to make Mysterious things plain; to explode and disuse all insignificant and puzzling words; to improve and apply little small threads of Mathematics to vast uses; and yet not to neglect, the finest Consideration, even of Atoms, where the same is necessary. The which purposes of theirs, I venture to say, do as much differ (both as to difficulty and dignity) from what is commonly called Wit (and which takes with far the greater part of Mankind,) as the skill of Drawing and Painting a Cloud or Periwig doth from that of Designing or Painting many complicated Figures of Men and Beasts in some one Table, wherein each is perfectly to express some particular passion, and all standing together to contain the true and entire Spirit of the Story represented: For, in the latter, precise exactness is indispensible, whereas in the former, not only liberty always, but even extravagancy sometimes is not only tolerable, but laudable. And when I have said this; I withal say, that there is one Glory of the Sun, another of the Moon, and another of the Stars, which may all consist together, without destroying or maligning each other. And all these several Glories shine steadily in your Grace's Firmament. Being, I say, appointed to publish this Exercise, I have presumed to dedicate it to your Grace. First, because the Society have been pleased to order it to be published; (I dare not say, as approving it, but as committing it to Examination.) Secondly, because your Grace doth not only love the search of Truth, but did encourage Me 30 years ago as to Inquiries of this kind. For about that time in Paris, Mersennus, Gassendy, Mr. H●●● Monsieur Des Cartes, Monsieur Roberval, Monsieur Mydorge, and other famous men, all frequenting, and caressed by, your Grace and your memorable Brother, Sir Charles Cavendish, did countenance and influence my Studies, as well by their Conversation as their Public Lectures and Writings: Much of which honours and helps I owe unto your Grace, and have a fresh remembrance of them. Thirdly, because my Lord Ogle being now about to carve a significant Figure upon my Lord his Son, by his careful Education of him, I thought it a service to his Lordship, as well as an Expression of my Thanks for his former acceptance of my Endeavours, to call upon him, not only to instruct my Lord his Son in some Mathematicsk, but also to store and stock him with variety of Matter, Data and Phaenomena, whereupon to exercise the same; since Lines & Numbers, without those, are but like Lute-strings without a Lute or a Hand. For, my Lord, there is a Political Arithmetic, and a Geometrical justice to be yet further cultivated in the World; the Errors and Defects whereof, neither Wit, Rhetoric, nor Interest can more than palliate, never cure. For, Falsity, Disproportion, and Inconsistence cannot be rectified by any sermocinations, though made all of figurate and measured periods, pronounced in Tune and Cadence, through the most advantageous organs; much less by Grandisonous or Euphonical Nonsense, farded with formality; no more than vicious Wines can be remedied with Brandy and Honey, or ill Cookery with enormous proportions of Spice and Sugar: Nam Res nolunt malè administrari. These are the Reasons, why I have put your Grace's Name to this Treatise; though there is a contrary reason, why it should have wholly shunned your Grace's sight and knowledge: which is, That your Grace might not perceive how little progress I have made in thirty years' time upon those Studies. However I hope your Grace will take what I have done for an Argument of my patience and perseverance in these pleasant, though profitless, Employments, and see, that no heterogeneous Cares and Troubles have or can quench my affections to Philosophy, as no distances of Time or Place have made Me less than formerly, Your GRACES▪ Most humble, most faithful, and most obedient Servant, WILLIAM PETTY. Ult. Decemb. 1674. To the Right Honourable WILLIAM Lord Viscount Brouncker, PRECEDENT OF THE Royal Society. My Lord, THE Observations on the Bills of Mortality were distinctly Dedicated to a Peer of this Realm, and also to the Precedent of the Royal Society, and both with good acceptance: Wherefore I have also (like the Author of those Observations) Dedicated this Discourse to his Grace the Duke of Newcastle, for the reasons in the foregoing Epistle mentioned; and I now again Dedicate the same to your Lordship. First, In Gratitude for the several assistances I had from your Lordship towards the Experiments mentioned in this Discourse. Secondly, Because your Lordship is an Eminent judge in those Matters, a Person whose Animadversions I shall take for Kindnesses; and who is able to excuse the Errors, and defend the Truths I have delivered. Lastly, For that near half the whole Discourse relates to Shipping, Artillery, Fortresses, Sea-banks, etc. which all concern his Majesty's Service, and part whereof are happily entrusted by him to your Lordship's Care; I thought I might express My affection to those his Majesty's Concernments even by offering this my Mite unto them. Upon the whole Matter, I have laid hold on this Occasion, to Publish my desire of being esteemed, My LORD, Your Lordship's most humble and faithful Servant WILLIAM PETTY. Ult. Decemb. 1674. ERRATA. PAge 6. l. 5. r. Proportion. p. 44. l. 1. r. be for being. p. 49. l. 6. r. etc. be. p. 49. l. 13. r. moreover for viz. ibid. l. 14. r. Mice, or rather some small Animals (whose correspondent parts are but 1/12 in length of the Horses.) ibid. l. ult. r. 12/144 for 1/144 p. 87. l. 10. r. Numerus for numerous. ib. l. 11. v. of for or. p. 88 l. 8. r. whereof for thereof. A DISCOURSE TO THE Royal Society. FOrasmuch as this Society has been censured (though without much cause) for spending too much time in matters not directly tending to profit and palpable Advantages (as the Weighing of Air and the like) I have therefore, to straighten this crooked stick, bent it and my present Discourse the quite contrary way, viz. to the Sails and Shapes of Ships; to Carpentry and Carriages; to Mills, Mill-dams, Bulwarks; to the Labour of Horses, and to several other particulars: The which are not only gross enough of themselves, but are also as grossly handled in this Exercise, to prevent the further imputation of needless Nicety, and to leave room for your own further thoughts upon the same. And forasmuch as We have been also complained of for producing nothing New, I have together with my Instances and Applications, above and hereafter mentioned, presented you as an Appendix, to what is said of Springs and other Elastique bodies, with a new Theory (as I think) of Elasticity itself, and that mechanically explicated in order to make a breach on this hard Rock in Philosophy, and to chip off a little of that Block which has long lain thwart Us, in the way of Our Inquiries. Upon the whole matter I have followed the Example of Elderly Divines, who finding their Flocks not to mend their lives by perplexed Discourses about Predestination, Transubstantiation, etc. betake themselves at last to preach Faith and Good Wooks, Neighbourly Love and Charity, or Doing as we would be done unto, and the like. For I have in this Exercise declined all Speculations not tending to practice, and ventured at few new Hypotheses, but that of Elasticity; rather calling upon you to review your own former Observations, and to apply your Mathematics to Matter, so as both may be improved to the profitable purposes hereafter mentioned. Wherefore the Title and Scope of this Exercise is, Several Instances, wherein the consideration of Duplicate & Subduplicate propoortion, or wherein the consideration of Sides and their Squares is of use in humane affairs. And the Instances which I have pitched upon for this day are these following, viz. 1. In the Drawing or Driving powers, which force Ships or other bodies through the water, with reference to the respective Velocities caused thereby. 2. In the shapes or sharpness of bodies, cutting or dividing the water, through which they are driven or drawn, and in the different Velocities arising from thence, where the Bodies and Forces are equal. 3. In the Strength of Timbers or other homogeneous materials applied to Buildings, to Carts, or any other Machinaments intended for strength: And how by a Model to judge the sufficiency of such Engine as is represented by it. 4. In the effect of Oars upon equal and like Vessels, according to their Numbers, Length, Blades, and Motions with or against the stream of smooth or uneven waters. 5. In the Motion or Travelling of Horses, on their several Paces, and with different Burdens on them. 6. In the Strength and Velocity of Mills and their Wheels. 7. In the Effects of Gunpowder. 8. In the Distance at which Sounds may be heard. 9 In the Distances at which Odoriferous▪ matters may be smelled. 10. In the Distance at which the Objects of Sight may be seen. 11. In the time of the Returns made by vibrating Pendules. 12. In the Lives of men and their Duration. 13. In Musical & Sounding Bodies, such as Strings and Bells. 14. In the Effects and Motions of Fire, and burning Spirits. 15. In the Rising and Falling of Bodies, but especially of Water in Pumps, Overshot Mills, Leaks in Ships, the Heights of Rivers at their head above their fall into the Sea. 16. In Bellows, 17. In the Prices of several Commodities, as Masts, Diamonds, large Timber, Amber, Lodestones, etc. 18. In Mill-dams, Sea-banks, and in the Bulwarks or Walls of Fortresses. 19 In the Compression of Wool, and other Elastic Bodies, and of the Air within diving Vessels, as also in the Effects of Skrew-presses upon several Materials. Having thus enumerated my several Instances, wherein Duplicate, and Subduplicate proportion is of great importance; I might now fall downright upon the Application of those proportions to each of the respective matters above mentioned. But because Custom hath made it almost necessary to make a Preface to every Discourse, my Preface to this one Lecture shall be such, as may serve me for many more; that is, an Explication of what I myself (at least) understand by Matter, Body, Figure, Place, Motion, Quantity, Quality, Habit, Time, Proportion, Weight, Swiftness, Force, and Elasticity; which I shall do without imposing or scarce recommending the same to any other. For I would be glad, when any man speaks to me in matters of importance, by words which he uses often, that he would first give me a Dictionary of such words, to contain what he himself meaneth by each of them. Wherefore I shall, as a Preface, prefix this Dictionary, wherein I dare not define Matter by Ens, or Substance, because I think most men conceive Matter better than they do either of these two words, Ens, or Substance. Nor do I define the words, Think, Consider, or Conceive, by the words, Soul, Spirit, Act, or the like, for the same reason. But presuming you all understand, conceive, imagine, or fancy the words Matter and Thought, as well as any other I can use, I venture to say as followeth, and first, That 1. Place is the Image or Fancy of Matter, or Matter considered. 2. Quantity, the Fancy of Place. 3. Ratio, several Quantities considered together. 4. Proportion, several like Rationes. 5. Situation, several Places considered together. 6. Figure is Quantity and Situation considered together. 7. Body is Matter and Figure considered together. 8. Motion is change of Place. 9 Time, the Image of Motion. 10. Quality, several Motions considered together. 11. Habit, the same Motions repeated. 12. Likeness, several Figures, or Qualities, and Proportions considered together. 13. Swiftness, Time and Place or Space considered together. 14. Force is Body and Swiftness considered together. 15. Right is the Image of Possession, and is to it as Place to Body. 16. Elasticity I shall speak of hereafter. In the next place, I suppose all the First Matter of the World to be Atoms; that is, Matter Immutable in Magnitude and Figure. I suppose Corpuscles to be as many Atoms joined together, as make up a visible or sensible Object, and that all juncture of Atoms is made by their Innate motions. Moreover I suppose, That every Atom is like the Earth's Globe or Magnet, wherein are three Points considerable, viz. two in the surface, called Poles, and one within the substance, called Centre, or rather Bias, because in Atoms we consider neither Magnitude nor Gravity. These Atoms also may have each of them such Motions as Copernicus' attributes to the Earth, or more. Lastly, Motion to or from a Point makes a straight Line, and, about it, a Circle. But from the Centre to several Points in the Circle, is Angle. We further say, that the motions of Corpuscles are compounded of the abovementioned motions of Atoms; and the motions of bigger and Tangible Bodies (viz. their qualities) are decompounded out of the Motions, Situation, Figure, and Magnitude of Corpuscles; and that out of, and by, the premises all Phaenomena in nature must be solved. And this is all the Preface I shall trouble you with, being (as was said) the Dictionary wherein to find what I mean by every material word I intent to use in this ensuing Exercise, which we thus begin, viz. The First Instance, Wherein Duplicate, and Subduplicate Ratio or Proportion is considerable, Is IN the Velocities of two equal and like Ships; which Velocities, I say, are the square Roots of the Powers which either drive or draw them; as, for example, Such two Ships having sails near double to each other, or as 49 to 25, the Velocity will be as 5, the square Root of 25 unto 7, the like Root of 49. Again, if the sails be near triple, or as 49 to 16, there the Velocity shall be as 7 (the Root of 49) to 4 (the Root of 16.) So as a quadruple Sail is requisite to double swiftness, and noncuple to triple; that is, The sails must be in duplicate proportion to the swiftness of the Ship; or this, in subduplicate to that. Again, let there be two Ships of Equal sails, but of unlike or unequal sharpness, suppose the head of one extremely obtuse or quite flat, and the head of the other to be an Isosceles Triangle added thereunto; I say, the swiftness of these Bodies shall be as the Roots of the Perpendicular of that Triangle to the Root of half the Base, or half breadth of the same. Secondly, Or if the same Triangular head be cyphered away into an Angle from bottom to top; then, as the Root of the same Perpendicular is to the Root of the Depth or Thickness, so are the Velocities. Thirdly, If the said head be cyphered both ways together, than the Proportion of Velocities shall be as half of one of the above mentioned Proportions added to the other whole Proportion: Ex. gr. Suppose the Perpendicular of the triangle-head be 36, the half breadth 9, and the whole depth be 4; then the one Proportion shall be as 6, the Root of 36, to 3, the Root of 9: The half of which Proportion is as 6 to 6; and the other Proportion is as 6, the Root of 36, to 2, the Root of 4. Now add the Proportions of 6 to 6, to that of 6 to 2, the sum will be, as 36 to 12, or as 3 to 1. Fifthly, Suppose two Paralellepipedons of unequal heads or resistances, Ex. gr. as 8 to 5, or 64 to 40: And suppose the Sail on the bigger, to that on the lesser, to be as 9 to 4, or 72 to 32; then the Velocity of the bigger shall be to the Velocity of the lesser, as the Root of 45 is to the Root of 32. For if the Resistances be as 64 to 40; then, if the sail of the bigger to that of the less were proportionable to the Resistances, the sail of the less should be 45, whereas we suppose it but 32. Wherefore the Velocity shall be as the Root of 45, which is almost 7, to the Root of 32, which is about 5½, that is, as about 14 to 11. Memorandum, That wetting of Sails (by lessening the intersperst apertures between the threads of the Sail-cloth) doth make the Sail, as it were, bigger; which biggerness may be known and measured by the increase of the Ships velocity upon such wetting. For, if the Ship should move one tenth part quicker after wetting than before, we may conclude the Sails are swollen to the equivalent of about ⅕ part bigger; for 100 (whose Root is 10) exceeds 81, whose Root is 9, by about ⅕ of 100 By these ways the different Velocities, arising from the different Trim of the same Ship, may be also computed, the best Trim being that which makes least resistance, caeteris paribus. Now, having said thus much of the Effects of Sharpness and Sails, (the two principal causes of Velocity in shipping, and unto which all others may be referred;) I shall add, That the want of these two Advantages are the chief cause, why short, bluff, undermasted Vessels sail cheaper than others. For suppose two Ships▪ of equal burdens, but of unlike dimensions, the main Beam of the one being scarce ⅓ of the Keels length, and in the other, a full ⅕; I say first, that the Hull of the latter shall cost ⅓ part more than that of the former, and the advantage as to sailing shall be scarce ⅙ part. Again, suppose, the sharper could carry ½ as much sail more as the bluffer, whereof the advantage in sailing would be ⅙ part more, in all ⅓. Now, where the Sails are as 2 to 3, the Masts and Yards must be as 4 to 9 in substance; and in value much more: And where the Masts and Yards are as 4 to 9 in weight and bulk, the Cordage and Rigging must be answerable: And where the Masts, Yards, Sails, and Rigging are great, the Wind-taught of the Ship will correspond, and will require proportionable Cables; and the weight of the Anchor must follow the size of the Cable, and the number of hands must be proportionable to all the premises: So as the one Ship will cost at least double as much as the other, and will sail at double charge of Wages and Victuals, Ware and Tear, etc. Now if no trading Ship be (one time with another) above 1/10 of her whole reign under sail, or 6 days in 60, suppose the sharper and larger-sailed Ship sail in 4 days what the other performs in 6; the difference will be but 2 days in 60, or 1/30 part of the Wages, and Victuals, and other charges; whereas the charges is supposed to be more than double. I say, this consideration is of great weight in Vessels of burden, especially such as carry gross and cheap bulky Commodities, neither liable to damage or perishing: Of which goods 7 parts of 10 of all Seacarriage do consist. But on the other hand, where safety against Enemies, speedy dispatch upon important occasions, or preoccupation of a Market are in the case, there sharpness and great Sails may be admitted to the greatest proportions practicable. Having thus digressed, I mind you that we said, Velocities are the Roots of Resistances and Extent of Sails, etc. It may be well asked, How we know the same, since that very few Seamen or Shipwrights, either in their writing or discourses seem to understand or own this important Position. To which I answer, that I have by many Observations, Calculations, and Comparisons, found the same to be praeter propter true, although there be many circumstances which intermingle themselves in this Experiment, so as to disturb and confound it: As namely, The ill placing of Masts, The ill cutting and standing of Sails, The ill Trim of the Vessel, with the Cleanness or Foulness of the same; The Sails more or less worn or wet; as also taught or slack Rigging, etc. Wherefore not only to avoid these last mentioned Intricacies, but also to make these Positions Examinable by every one that desires it; I say, that the different Velocity of Bodies (of several sharpnesses, and as drawn or driven by different Powers of knocks or falling weights,) have been by myself and others much experimented in large Canales, or Troughs of water, fitted with a convenient Apparatus for that purpose, and by no man more, nor more judiciously, than by the Right Honourable the Lord Brouncker, Precedent of this Society. For I do not think it hard to conceive, that Weights and Sails are powers of like Effect, and reducible to the same Principle; so as if a Body have moved in double velocity, when drawn by a quadruple weight; and in triple, when by a noncuple weight; I doubt not but the same will hold in Sails, or other impellent Powers of the same proportions. And for the further clearing or easier trying hereof, I offer two small Machinaments heretofore made in this Society: The one, to measure the Velocity of the Wind, and the other its Power or Equivalency to Weight; whereby it did and will appear, when the wind is of double velocity, it will stir a quadruple weight; and the like in other cases according to the proportions of Roots and Squares above mentioned. The same may also be seen even in any good Turnspit-Jack, where a quadruple weight makes double Velocity (at the same distances of Time from the beginning of the Motion) both in the time of the Weights descent, as also in the Revolutions of the Fly, and each intermediate Wheel. Now perhaps the reason of these Phaenomena may be here expected; to which I answer, that the many parallel Instances above and hereafter mentioned, do, like concurrent witnesses, prove the premises, at least as to any practical use. And as for giving other reasons (which I take to be Explaining this Subject from the very first Principles of Atomical Matter, and Motion) I leave it to discourse, as too long for this Exercise. The Second Instance Is in the Strength of Timber, etc. LEt there be Square Rods or Pieces made of any Clean Timber, or other Materials, whose Ends let be supported with convenient Blocks or Fulcra: These Rods in Experience will bear weight hung in the middle of them, according to the proportion of their lengths or distance, between the Fulcra; that is to say, a Rod A. being of double length to the Rod B. will bear ½ the weight which B can bear; and being of triple length, it will bear one third; & sic de caeteris. Again, let two of those equal and alike square Rods be placed one upon the other (so as to touch and sit,) then the two together shall bear 4 times as much as one alone, and three of them, placed as aforesaid, shall bear nine times as much, and so on in proportion of Roots to Squares. Again, lay the same two Rods side by side, to each other, than they shall bear but double, three shall bear triple, and so forward, in Arithmetical proportion. From whence it follows, that four of them placed square, shall bear eight times as much as one alone. But if the same four Rods taken as One, being of double length making an Octuple quantity to One, they shall bear but four times the weight of One alone. So as two like pieces of Timber, that are in cubical or triplicate proportion of their Sides, are strong but according to duplicate proportion, or the Squares of their respective Sides; and consequently, to have like Vessels (differing in Content as the Cubes of their like Sides) equally strong, the Timber of which they consist must be Quadratoquadratic; that is to say, a Ship of 400 Tuns, equally strong with one of 50, must have not only 8 times as much Timber in it, but 16 times; which is seldom or never done. Which defect is the true Reason, why great Shipping is both Dearer and Weaker than small Shipping, (no Ship in the world being so strong as a Nutshell;) I say, Weaker, for what is here said; and Dearer, for what shall be said hereafter in the sixteenth Instance of Masts, Diamonds, etc. And on the other hand, if the Timbers were Quadratoquadratic, than the Ship of 400 Tuns would be loaden with her own Materials; if the Ship of 50 Tuns were not over-timbered. Now, for not well understanding these matters, many men designing Engines of strength, do make Models of such Machinaments by a Scale (suppose wherein an inch represents a foot,) by which the Model is the 1/1728 part of the Engine intended: And thereupon they conceive, that if the Model be strong enough to bear 1/1728 part of what the great Machinament is intended to bear, that then the said great Machinament will be strong enough. Whereas indeed the Model must bear the full 1/144 of what is intended for the great Machinament; otherwise great mischiefs will appear in the Work. Wherefore the Square of the Linear Difference between the Model and Engine, is the measure and way of trying the strength and sufficiency sought for: The ignorance whereof hath made many a poor Projector. Upon these Principles, a Cask which will hold a Tun, aught to have 16 times as much Timber in it, as the Cask which holds only a Barrel, or ⅛ of a Tun; provided one be as strong as the other (which is not usually seen.) For the bigger Vessels, Carts, etc. they are usually the weaker compared with the strength of the lesser; which appears also in Animals, whose strength is as the Square Roots of their weights and substance, viz. if 1728. Mice were equiponderate to one Horse, the said Horse is but 1/144 part as strong as all the said Mice. From these considerations the Scantlings of Timber in Buildings must be adjusted; as for example, Let the Walls of any Room be infinitely, that is, sufficiently strong; let the length and the breadth of the Room be given: Next, suppose the Room is to be made so strong, as that every foot and a half square shall bear a Man, and so, that 31½ square feet should bear a Tun weight, (reckoning 14 men to the Tun:) Last, let the strength of the Timber be also given. Now the Questions are, to find the Scantlings of the Girders, Gise, etc. first in square pieces, and afterwards by altering the Squares into more advantageous ablong Sizes; as for example, Let the Room be supposed 26 foot long and 20 broad, viz. 520 foot in the Area, and able to receive about 250 men, and to bear about 16 Tuns. Suppose the Timber be such, as whereof a Rod of an inch square, and 20 foot long, will bear 1/20 part of an hundred weight; or, that 20 such Rods, or a Board of 20 inches broad, and 20 foot long within the walls, an whole hundred weight; and so the whole Floor consisting of about 16 such Board's, but 1600. Now if the same Board were plank of 4 inches thick, it would bear 16 times 1600 or 256 hundred weight: If 5 inches, 400 hundred weight: But the whole weight designed being but 325 hundred, some size between 4 and 5 inches thick will suffice in this case, where we suppose the Floor to be of plank without Gise or Girder. Next, suppose instead of this Planck there be used Gise of double thickness to the said Planck, and placed at quadruple distance; I say, the Effect and Strength will be the same with half the stuff. And I also say, that one Girder alone of 18 inches square, and 20 foot long, is near Equivalent to the 17 Gises of 9 inches deep, and 4½ broad-abovementioned; which Girder has but half the stuff which the Gise had; as the Gise did contain but half the stuff, which the 4½ inch-Planck first mentioned did contain. Which saving of stuff is the reason of dividing Plank into Girders, Gise, and Board. Where note, that these Proportions and Scantlings are not offered as exact and best for practice, but only to intimate the method of enquiring into these matters so useful in the world. The Third Instance▪ In the Oars of a Boat, etc. TO determine or make a good estimate of the power of Oars, I first, for easier calculation, suppose a Paralellipipedon-Boat or Vessel, of breadth fit for a pair of Skulls, viz. of about 5 foot broad, and of length sufficient for 9 such Skulls or Oars, viz. about 30 foot long, and one foot deep, and to draw but three inches water. Next, I suppose, that every Scholar with his Skulls and Bench, etc. their weight to be equivalent to three Cubical foot of water; so as every pair of Skulls (with its appurtenances) depresses or sinks the Vessel 1/50 of a foot, or about ¼ of an inch. Now, suppose also a smooth calm standing water, in which one Rower will row this Vessel 12000 foot, or above two miles in an hour or 3600 seconds; I say then, that, if one Remex or Scholar move 12 quarters or 3 inch. es draught, 12000 feet forward in 3600 seconds; then 4 like Rowers shall move the same Vessel, drawing 15 quarters, or 3¾ inches of water, the same 12000 feet, in 1800 seconds plus 360 seconds, or in all, 2160 seconds: And that 9 shall row the same Vessel, as the Root of 21 to the Root of 208, which is, as near 3 to 7; or in 3/7 of the time that one Rower alone could have done the same. Again, suppose each Oar lengthened from two to three, and that as many strokes are made in the same time as before; then the Velocity shall increase proportionably. But suppose, that the Oars remain of the same length, but that the Blade be doubled; then the Velocity shall increase but according to the Roots of that doubling, or as 10 to 7, or 7 to 5, etc. supposing still the same number of strokes, within the same time, in every Case or Experiment. Again, suppose these Experiments be made not in still water, but in water which runs 6000 foot an hour; then, against the stream the Velocity will be lessened by one half, and accelerated answerably with it. Lastly, if the said water be so rough, as that the Vessel heaus and sets, suppose 20 degrees of the Quadrant in it; then, for as much as the Boats way will be increased as much as the Tangent of 20 degrees exceeds the Radius, the way or Velocity of the Boat must abate proportionably. The Fourth Instance. In the Motion of Horses. SUppose an Horse can travel 5 miles an hour with 200 pound burden on his back; then with half the said burden he shall travel 7; and with double but three miles and a half. Again, suppose a Horse with 200 pound burden can endure to travel 10 hours per diem; then with half the same burden he may endure 14 hours, and with double but 7 hours. Lastly, suppose a Horse (as Race-horses) can run after the rate of four miles in ⅛ of an hour, or 32 miles per hour, than they can run about 6 miles 1/28 in ¼, or after the rate of 24 1/7 miles per hour; and in one half an hour can run 8 miles, or after the rate of 16 miles per hour; and in a whole hour can run 12 1/14 miles; and in 2 hours can run 16 miles, or 8 miles per hour; and in 4 hours can run 24 miles, at 6 miles per hour; and in 8 hours 32, or 4 miles per hour; and in 16 hours may go 48 miles, or 3 miles in an hour. All which agrees well enough with Experience. The Fifth Instance, In Mills. WHere the wind blows, suppose, on a Saw-mill, in double Velocity, there the Saw-mill, which carried but one Saw, shall carry four; If treble, shall carry nine. And the like is true of water gushing out upon the floats of Under-shot Mills; as may be seen in the Stampers of Paper-Mills, the Stocks of Fulling-Mills; and other Works of the like nature. The Sixth Instance, In Gunpowder. THe way of a Bullet, shot out of a good Gun, is as the square Roots of the quantity of the Gunpowder fired; I say, of Powder fired, because what goes out unburnt, goes rather as Shot than Powder; and the Length of Guns signifies only the constraining of the Powder within the Lines of Direction, till it be all fired: The use of hard ramming and screwing of Guns, being also the same; and the excellency of Powder being to fire quick, and before it goes out of the Gun. I say therefore, the Velocities caused by Gunpowder are as the Roots of the Powder fired, that is to say, 4 pound of Powder, all equally fired within the Piece, shall carry a Bullet twice as far as one pound shall do; and in Time, as 10 to 7; which last mentioned numbers are the Roots of the double distances aforementioned. Now, if the Capacity of the Concave of Guns ought to be, as the Weight of their Bullets or Powder; then, if the just length of any one Gun hath been well found by good Experimentation, then may also be known the length of every Gun for every Bullet respectively. As, for example, suppose a Gun, that carries a Ball of 5 inches Diameter, be 10 foot long in the Concave, than the Content of the said Concave will be 3000 Cylindrical inches. Now the question is, how long must the Piece be, which carries a Bullet of 7 inches Diameter? I say, that forasmuch as the Weight of the 5 inch Bullet, to that of 7, is as 125 to 343; the Concave of the greater Gun must be in the same proportion to 3000, viz. 8232 like inches, so as it may contain and fire a proportionable quantity of powder: Which 8232 being divided by the Area of the Bullet, 49, the Quotient will be 168 inches, or 14 foot; that is (to speak shortly and plainly) The Length of Guns must be measured by the Diameters of their respective Bullets. I cannot say, I have tried the effects of Gunpowder to be in the abovementioned proportion, but have credibly heard it to be so; and because of the Similitude of Sails, Weights, Knocks, and the other points above described, unto this of Gunpowder, I believe it; and recommend it to your further thoughts and experience. The Seventh Instance. Of Sounds. LEt there be many Equal Sounds; I say, that the Distances, at which they may be heard, are the Roots of the Numbers of such Sounds. For, four Muskets will be heard twice as far as one, and nine thrice; and so of the rest. By which reckoning, the hearing of some of our Fleets Engagement with the Dutch even to S. James' Park near this City is easily solved; and the truth of that Observation doth reciprocally countenance this Doctrine. For suppose both Fleets (consisting of two hundred Ships great and small) had about 12000 pieces of Ordnance on board them, which at a Medium suppose to be Demi-Culverins: Suppose also, that a Demiculverin, with the same circumstances of Wind and Air, may be as easily heard five miles, as the said Engagements were heard 160 miles. Then I say, that 1024 of the said 12000 Guns firing together, or very near the same time, might (as they were) be well heard 160 miles; and that about 4000 such Guns might as well be heard 300- miles, as one Demi-Culverin five miles; which last point I add, to prevent the unbelief of a probable matter, when it shall happen. Now what effect this had in the Popes Presage of the Battle of Lepanto, I know not. The Eighth Instance Of Smells I Say the same of Smells, viz. that the Distances at which they are perceived are the Roots of the Quantity of the Matter out of which they are emitted; which Doctrine I apply to solve what I once did hardly believe, viz. that Ships coming from America towards Portugal, did smell the Rosemary and other odoriferous herbs 60 miles off from the Land: The which seems not only credible, but very likely. For, if a foot square of a Rosemary-Field may be smelled one Perch or Rod (whereof 320 make a mile,) then about 8000 Acres of Land, whereon such scented Plants do grow (or a piece of Land about 4 miles long, and 3 miles broad; or 6 miles long, and 2 miles broad) may be smelled 64 miles: And 72000 Acres of the like Land, or a parcel of such Land about 11 miles square, may be smelled as many leagues, or near 200 miles. And this Consideration I pitch upon, as one of the grounds whereupon I would build a Doctrine concerning the Influence of the Stars, and other Celestial or remote Bodies upon the Globe of the Earth, and its Inhabitants, both Men and Brutes. The Ninth Instance Concerns Visible Objects. I Say also, that four equal and like Candles will give light but twice as far as one, and 9, thrice as far; and that 16 will also enlighten but 4 times as far as one, etc. And if a Flag or Ships-Vane of a yard square may be seen a league off at Sea, it must be 2 yards square, or 4 square yards to be seen 2 leagues, and so forward. But whoever will make experiment hereof, must first consider, how many miles in thickness of a Middling, Clear, and Diaphanous Air do make an Opaque. For we find, that although a very thin plate of clear Glass seems to hinder our sight of near Objects but very little; yet we also know, that great number of them (suppose one hundred) can scarce be seen through at all. Hereunto also must be added the Consideration of the Convexity of the Earth; and then I doubt not, but this Doctrine (of Roots and Squares) rectified and corrected with the two additional Considerations last mentioned, will hold concerning Visible and Lucid Bodies, as was above propounded. The Tenth Instance, In the Time of the Vibration of Pendules. THe times in which the Returns of a Vibrating Pendulum are made, are the Roots of the Distances between the Centre of the Pendulum, and the Centre upon which it moves. I shall need to make no application of this Truth, since we all enjoy the benefit of it in our more regulated Clocks and Measures of Time, which are now in common use, and from whose Improvements we may most hopefully expect a better measure of Longitude upon the Surface of the Earth. The further uses which may be made hereof, (it being a very simple and examinable Experiment) is to witness and give evidence to other the more abstruse and complicate Positions, which are of the like and parallel Nature. The Eleventh Instance In the Life of Man, and its Duration. IT is found by Experience, that there are more persons living of between 16 and 26 years old, than of any other Age or Decade of years in the whole life of Man (which David and Experience say to be between 70 and 80 years:) The reasons whereof are not abstruse, viz. because those of 16 have passed the danger of Teeth, Convulsions, Worms, Rickets, Measles, and Smallpox for the most part: And for that those of 26. are scarce come to the Gout, Stone, Dropsy, Palsies, Lethargies, Apoplexies, and other Infirmities of Old Age. Now whether these be sufficient reasons, is not the present Enquiry; but taking the aforementioned Assertion to be true; I say, that the Roots of every number of men's Ages under 16 (whose Root is 4) compared with the said number 4, doth show the proportion of the likelihood of such men's reaching 70 years of Age. As for example; 'Tis 4 times more likely, that one of 16 years old should live to 70, than a newborn Babe. 'Tis three times more likely, that one of 9 years old should attain the said age of 70, than the said Infant. Moreover, 'tis twice as likely, that one of 16 should reach that Age, as that one of 4 years old should do it; and one third more likely, than for one of nine. On the other hand, 'tis 5 to 4, that one of 26 years old will die before one of 16; and 6 to 5, that one of 36 will die before one of 26; and 3 to 2, that the same person of 36 shall die before him of 16: And so forward according to the Roots of any other year of the declining Age compared with a number between 4 and 5, which is the Root of 21, the most hopeful year for Longaevity, as the mean between 16 and 26; and is the year of perfection, according to the sense of Our Law, and the Age for whose life a Lease is most valuable. To prove all which, I can produce the accounts of every Man, Woman, and Child, within a certain Parish of above 330 Souls; all which particular Ages being cast up, and added together, and the Sum divided by the whole number of Souls, made the Quotient between 15 and 16; which I call (if it be Constant or Uniform) the Age of that Parish, or numerous Index or Longaevity there. Many of which Indices for several times and places, would make an useful Scale of Salubrity for those places; and a better Judge of Airs than the conjectural Notions we commonly read and talk of. And such a Scale the King might as easily make for all his Dominions, as I did this for this one Parish. The Twelfth Instance In Music. TAke a Musical String, one end thereof being fastened; hang unto the other (over a convenient Bridg) any weight which may strain it to some grave Musical Tone or Note; then set some other string of near the same length, Unisone thereunto. Lastly, instead of the first weight, hang to the first String the Quadruple of the same weight; and it will appear, that the String with the quadruple weight shall yield a Tone of an 8th or Diapason above itself, when singly charged. The reason is, because the quadruple weight doubles the number of Vibrations, (2 being the Root of 4:) And for that the Ratio Formalis of Tones lieth in the number of the Vibrations; and of the Diapasons, in the doublness of such numbers. By the same Method of hanging-on several weights at one end of the same String, all Tones may be produced, of which such String is capable. The Tones or Notes also of like Bells and Drums do follow the same proportions of their Tension and Metal, so as able Artists can cast Bells in Tones assigned. The Thirteenth Instance, of Fire and Spirits. LEt a Cylindrical Flatbottom Vessel be filled with Water, and let it be tried, in what time one Lamp or Candle would make the water boil through, or come up to its greatest heat: Then see, in how much lesser time, 2, 3, or 4 more like fires will hasten the same effect. I cannot speak positively hereof, but know from several Observations, that the Acceleration abovesaid shall not be made in Arithmetical Proportion; forasmuch as I know, that in Fireworks great Fires are more profitable than small; as in Brewer's Coppers, and Iron-works may be seen; wherein double Fires produce more than double dispatch or advantage. I shall therefore suspend this matter, and pass to the measuring of the Spirituosity of Liquors, or in what proportions several Liquors contain more or less of inflameable or ardent parts. Now in this case I conceive, the Consideration of Roots and Squares is also material; for I understand by strength or multitude of Spirits, the Space, greater or lesser, into which such Liquors will be rarified, or will fill with Spirits: As for example, if a Pint of Water rarified into Vapour will fill a Globe but of 3 foot Diameter; and a Pint of rectified Spirit of Wine will fill a Globe of six foot diameter, or 8 times as large as that of Water; I shall say, that there is 8 times as much Spirit or Vapour in one as in the other. But if these Liquors were put into open Lamps or Vessels, there the space in which the Spirits rise, are the Roots, whose Squares do show the Spirituosity of those Liquors: Ex. gr. Let there be a Lamplike Vessel of common Aquavitae; in which place a Week as high as the same will burn by the rising of the Spirit unto it, suppose an inch above the surface of the Liquor: Now, let there be a like Equal vessel with such a Spirit, as will rise up higher, suppose to a Week placed two inches above the Surface; in this case, I say, that the latter Liquor is quadruple in strength or extent of Spirit to the former; for 'tis certain, that as the Spirit riseth double upwards, so also it emitteth or rarifieth itself double also sideways; and consequently the quantity of the Spirit or Vapour must be quadruple; and so of other proportions. The Fourteenth Instance, of Rising and Falling Bodies; but particularly of Waters in Pumps and River-streams. LEt it be observed in the Transparent Pipe of a Forcing Pump, at how many strokes the Water is forced from the Bottom to the Top; and let as many marks be made at the several places unto which the Water mounted at every stroke (which strokes we suppose to be all in Equal Times;) it will appear, that all the said Divisions will be according to the Proportions or the Logarithms abovementioned. As for the Descents and Accelerations of falling Bodies, the Times are the Roots of these Spaces, which they fall in the said times respectively. The great effect whereof we see in Overshot-Mills, where a little water falling upon a Wheel of a large Diameter, produceth wonderful Effects; the which may be well computed upon the Principles we hold forth. Waters also have greater forces in the abovementioned proportions, as the hole or place whereat they issue is lower from their Surface; as may be seen in all Breast▪ and Undershot-Mills; where it is pleasant to divide the Sinking of the water into Equal Spaces, and to count the Clacks, Revolutions or Strokes made within the Time of the waters sinking every such equal Space; for therein the abovementioned Logarithmes may also be observed. Unto this head may be referred the Leakage of Ships. For let there be a hole in a Ship somewhere under water; then let it be seen, what water comes in at the said hole, within any space of Time; then let the like hole be made at double the perpendicular distance from the top of the water, and there shall come in four times as much as at the upper hole; and let a third be at three distances, and that shall admit 9 times as much, etc. Again, let there be two Equal holes or Leaks in a Ship, the one at Head, and the other at Stern, and let the Ship be in motion; then the Leakage at the Head is composed of the pressure of the water from the Surface, and of the Ships Motion together. Moreover, if the Ship make double way, the Leakage will be quadruple; if treble way, non cuple, etc. Wherefore to stop Leaks afore, the Ship must stop its motion, lie by, or bear up to go with the Wind and Sea, etc. Lastly, I shall add, that the Swiftnesses of Waters or River-streams, are the Roots of the Power that causes them; which causes are Steepness or Descent in a sharper Angle from the Perpendicular. Wherefore knowing by observations, what degree of Steepness causeth any degree of Swiftness; hereby, and by our Doctrine, the Height of ground where any River riseth above its fall into the Sea, may be computed. The Fifteenth Instance, In the Blast of Bellows. IN Ironwork Furnaces are the greatest and most regular moving Bellows that are any where used; the which are commonly turned by the evenest overshot Wheels. Now the Times wherein these Bellows rise and fall, are Roots of the Strength of such Bellows-blast upon the fire; for rising in double Quickness admits double air in the same Time; which being in like manner squeezed out again, double Quickness makes double Expulsion, and consequently double Swiftness; (the whole passing through the same Twire-pipe in half the time;) and double Swiftness makes quadruple effects upon the fire or Furnace, as aforesaid. The Sixteenth Instance, In the Price of several Commodities. SUppose a Mast for a small Ship be of 10 inches Diameter, and as is usual, of 70 foot in height, and be worth 40 s; then a Mast of 20 inches through, and double length also, shall not only cost eight times as much, according to the Octuple quantity of Timber it contains, but shall cost 16 times as much, or 32 l. And by the same Rule, a Mast of 40 inches through shall cost 16 times 32 l. or 516 l. Of which last Case there have been some instances. But whereas it may be objected, That there are no Masts of four times 70, or 280 foot long, I still say, that the Rule holds in common practice and dealing. For, if a Mast of 10 inches thick, and 60 foot long, be worth 30 s; a Mast of 20 inches throughout, and 80 foot long, shall be worth 15 l. And a Mast of 40 inches thorough, and 100 foot long (not 280 foot) shall be worth near 100 l. Moreover, suppose Diamonds or Pearls be equal and like in their Figures, Waters, Colours, and Evenness, and differ only in their Weights and Magnitudes; I say, the Weights are but the Roots of their Prices, as in the Case aforegoing. So a Diamond of Decuple weight, is of Centuple value. The same may be said of Lookingglass-Plates. I might add, that the Loadstone A, if it take up 10 times more than the Loadstone B, may be also of Centuple value. Lastly, A Tun of extreme large Timber may be worth two Tuns of ordinary dimensions; which is the cause of the dearness of great Shipping above small; for the Hull of a Vessel of 40 Tuns may be worth but 3 l. per Tun, whereas the Hull of a Vessel of 1000 Tuns may be worth near 15 l. per Tun. From whence arises a Rule, how by any Ships Burden to know her worth by the Tun, with the Number and Size of her Ordnance, etc. The Seventeenth Instance, In Mill-Dams, Sea-Bancks, and Bulwarks of Fortresses. SUppose any Wall, Damn, or Bank, to be just sufficient to keep out or resist the Sea, or other Stream against the appulse of its waters, being of a certain force; I say, that to make this Wall or Dam strong enough against a double swiftness of appulse, it must be augmented by quadruple thickness; and if it must be made sufficient against the greatest violence which ever was observed, than that violence being known, is the Root of the number by which the Walls thickness must be augmented. So Cannon-Bullets do Execution or batter in duplicatâ ratione of their swiftness; and therefore Ramperts must be strong and thick in duplicatâ ratione of the said swiftness, which depends upon the Distance of the Battery, and the degrees of Tardation, which Bullets make in every part of their way between the Gun and the Rampert, which they are to batter. Where note, that Bullets commonly beat out a Cone of Wall, whose Vertex is in the Bullets Entry, and like the Conical Fovea to be seen in the Sand of an Hourglass. The Eighteenth Instance, In the Compression of Yielding and Elastic Bodies, as Wool, etc. SUppose some Cylindrical or other paralleled sided Vessel, filled with Wool, or Down, or Feathers, or other Elastic Materials; let the same be covered with a movable Head (such as in pressing of Pilchards they call a Buckler;) then first observe, how low the Buckler descendeth by its own weight; and then upon this Head or Buckler lay a triple weight, to make the whole quadruple, and it will appear, that the Buckler will sink but just as much lower; and being Noncuple, another like Space lower: So as the several Spaces of Depressions are the Roots of the depressing Powers. From hence may be seen, how the Force must be increased at every Turn or Thread of a Screw-Press; which being done according to the proportions here understood, I doubt not, but a Light Substance with a convenient Apparatus, might be compressed unto the Density and Weight even of Gold. But, that Silver might be so condensed, I made no question, till I heard of some Anomaly in the practice, which I must better consider of. The further Truth whereof doth appear in the Vnder-waterAir within the Vessels of Water-Divers, who the lower they go, do find their stock of Air more and more to shrink; and that according to the Roots of the Quantities of the super-incumbent Water or Weight. In like manner take a Bow, and hang any weight to the middle of its string, and observe how low it draweth the said string. Now, if you shall quadruple the same weight, it will draw down double the first distance, and noncuple will draw it down treble, etc. So as in a drawn Bow, let the Arrow be divided into quotcunque parts, each equal part of the Tension carrieth the Arrow to an Equal Distance, notwithstanding each equal part of the Tension was made by Unequal power, and that each equal Space or Part also of the Arrows first flight requires Unequal Force, viz. least strength at first, and most at last; and that, in the proportion first mentioned. So in the Fuze of a Watch, the greatest strength of the Spring is made to work upon the shortest Vectis; and the least upon the longest, so as to equalise the whole. The like also happens in the Traction of Muscles upon two Bones with a turning Joint between them; which Bones and Muscles make a Triangle, whereof the Muscle is the Base, subtending the Angle-Joynt. Now in the working, the Muscle is strongest, when the Vectis is smallest, as lying most obliquely; and vice versâ, when the Muscle and moving Bone come to make a right Angle. An Appendix OF ELASTICITY. HAving done with the Consideration of duplicate and subduplicate Proportion in Elastic Bodies and Materials, I hope it will not be amiss to subjoin a short Appendix of Elasticity itself, whereby to draw forth the better thoughts of other men for Countenance or Correction. Wherefore I say as followeth; viz. First, Supposing every Body to have a Figure or Positure of its own, out of which it may be disturbed by External Force; I say, that Elasticity is the power of recovering that Figure, upon removal of such Force. 2. I think it easiest to consider Elastic, Springing, or Resilient Bodies, as Laminae, Laths, or Lines; so as a straight Lath, being by force bend circularly, doth upon the removal of that Force, return to be straight again by its Elasticity; and a Circular Hoop being forced straight, leaps back into its own crookedness by its Elasticity. 3. Elastic Bodies in their returns do overshoot their own Natural Positure, and vibrate cis citrà the Point they seek, as doth a Pendulum, or MagneticNeedle, till at length they rest; the one in his Perpendicular, and the other in his Meridian. 4. An Elastic Body is a gross Tangible Body, which is made of Corpuscles, or the smallest Bodies that can possibly be seen; and these Corpuscles are made of Atoms, or the smallest bodies in Nature (such as whereof a Million doth not perhaps make one of the Corpuscles last mentioned.) 5. I know no reason, why we may not, upon occasion, suppose Atoms to be of several Figures and Magnitudes, provided we suppose them immutable, such as Corpuscles are not; gross tangible Bodies being very mutable by the various Additions and Detritions that befall them. 6. I suppose in every Atom three such points as we all see and know to be in the Globe of the Earth, and in every Magnet, viz. two Poles in its Superficies, and a Central point within its substance, which I call its Bias. The Heavens also visibly have their Poles, and must have a Centre of Gravity or Magnitude, or some other Central and predominant Point. 7. I suppose every Atom may move about his own Axis, and about other Atoms also, as the Moon does about the Earth; Venus and Mercury about the Sun; and the Satellites jovis about jupiter, etc. 8. I suppose, that the Bias of one Atom may have a tendency towards the Bias of another near it, and that the Byasses of many Atoms may tend to some common point without them; as we see in Electrical Bodies, and in the Globular drops of Water and Quicksilver, and all Mucilaginous Substances. 9 I suppose, that all Atoms have, like a Magnet, two Motions, one of Gravity, whereby it tendeth towards the Centre of the Earth, and the other of Verticity, by which it tendeth towards the Earths-Poles, and whereby Magnets join to each other by their Opposite Poles. 10. All Atoms by their Motion of Verticity or Polarity, would draw themselves, like Magnets, into a straight Line, by setting all their Axes in directum to each other; did not the Motion of their respective Byasses towards each other, and towards other Points, kerb them into a Triangle, whereof the Two Axes of Two Atoms are two sides, and the distance between the Bias of each making the third side: Wherefore I call the Polar Motion abovementioned, the Motion of Rectitude; and the Motion of the Biasses, the Motion of Angularity or Curvity, or the Angular or Curve Motion. 11. I suppose, that all these Motions may be of different Velocities, and that by Contra-colluctations they balance each other, sometime into seeming rest: I say, seeming, because perhaps there is no rest in Nature. Lastly, I might suppose (even without a Metaphor) that Atoms are also Male and Female, and the Active and Susceptive Principles of all things; and that the abovenamed Byasses are the Points of Coition: For, that Male and Female extend further than to Animals, is plain enough; the fall of Acorns into the ground, being the Coition of Oaks with the Earth. Nor is it absurd to think, that the words in Genesis, [Male and Female created he them] may begin to take effect, even in the smallest parts of the first Matter. For although the words were spoken only of Man; yet we see they certainly refer to other Animals, and to Vegetables in manner aforesaid, and consequently not improbably to all other Principles of Generation. Conclusion. To Conclude, I hope I may say, that these my Principles, are Principles indeed; for there can be no fewer nor easier than Matter and Motion. My Matter is so simple, as I take notice of nothing in each Atom, but of three such Points as are in the Heavens, the Earth, in Magnets, and in many other Bodies. Nor do I suppose any Motions, but what we see in the greater parts of the Universe, and in the parts of the Earth and Sea. Again, all the Motions I fancy in my Atoms, may be represented in gross Tangible Bodies, and consequently may be made intelligible and examinable. Moreover, I hope none of my Suppositions are inconsistent with each other, nor do necessarily infer any absurdity or falsehood. And lastly, I hope they solve all the Phaenomena of Elasticity, and, as I think, of Hardness, Fixedness, Tenacity, Fluidity, Heat, Moisture, Fermentation, and the rest. All which is humbly submitted to the Censure of this Society; whose Atoms or inseparable Members I wish may happily Conglomerate, and Unite themselves into the most fixed and most noble Bodies amongst the Sons of Men. FINIS.