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. ... Petty, William, Sir, 1623-1687. 1674 Approx. 69 KB of XML-encoded text transcribed from 83 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-03 (EEBO-TCP Phase 1). A54611 Wing P1919 ESTC R4342 12579411 ocm 12579411 63700 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A54611) Transcribed from: (Early English Books Online ; image set 63700) Images scanned from microfilm: (Early English books, 1641-1700 ; 328:9) 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. ... Petty, William, Sir, 1623-1687. [31], 135 p. Printed for John Martyn ..., London : 1674. "An appendix of elasticity" (p. 121-135) deals with theory of atomic structure. Errata: p. [30]. Reproduction of original in Yale University Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. 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Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Physics -- Early works to 1800. Atomic structure -- Early works to 1800. 2003-07 TCP Assigned for keying and markup 2003-07 Apex CoVantage Keyed and coded from ProQuest page images 2003-10 Mona Logarbo Sampled and proofread 2003-10 Mona Logarbo Text and markup reviewed and edited 2003-12 pfs Batch review (QC) and XML conversion 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 Iohn Martyn , Printer to the Royal Society , at the Bell in St. Pauls 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 , &c. 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 Mathematicks , by shewing 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 Chymerical 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 , 't is the Profession of the Society , to make Mysterious things plain ; to explode and disuse all insignificant and puzling words ; to improve and apply little small threds of Mathematicks 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 onely liberty always , but even extravagancy sometimes is not onely 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 steddily in your Graces 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 onely love the search of Truth , but did encourage Me 30 years ago as to Enquiries 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 Publick Lectures and Writings : Much of which honours and helps I ow 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 onely 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 Iustice 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 Nonsence , 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 Graces Name to this Treatise ; though there is a contrary reason , why it should have wholly shun'd your Graces sight and knowledg : 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 , PRESIDENT 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 President 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 Iudge 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 , &c. which all concern his Majesties Service , and part whereof are happily entrusted by him to your Lordships Care ; I thought I might express My affection to those his Majesties Concernments even by offering this my Mite unto them . Vpon the whole Matter , I have layd hold on this Occasion , to Publish my desire of being esteemed , My LORD , Your Lordships 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. &c. be . p. 49. l. 13. r. moreover for viz. ibid. l. 14. r. Mice , or rather some smal 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 streighten 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 grosly handled in this Exercise , to prevent the further imputation of needless Nicity , 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 it self , 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 Enquiries . 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 , &c. 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 Mathematicks 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 pitcht 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 judg 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 Burthens 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 smelt . 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 , Loadstones , &c. 18. In Mill-dams , Sea-banks , and in the Bulwarks or Walls of Fortresses . 19. In the Compression of Wooll , and other Elastick 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 down-right upon the Application of those proportions to each of the respective matters above mentioned . But because Custome 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 my self ( 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 contein 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 joyned together , as make up a visible or sensible Object , and that all Iuncture of Atomes is made by their Innate motions . Moreover I suppose , That every Atom is like the Earths Globe or Magnet , wherein are three Points considerable , viz. two in the surface , called Poles , and one within the substance , called Center , or rather Byas , 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 streight Line , and , about it , a Circle . But from the Center 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 premisses 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 intend 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 treble ; 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 wayes together , then 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 threds 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 burthens , but of unlike dimensions , the main Beam of the one being scarse ⅓ of the Keels length , and in the other , a full ⅕ th ; 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 premisses : 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 Tare , &c. 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-sail'd Ship sail in 4 dayes what the other performs in 6 ; the difference will be but 2 dayes 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 , &c. It may be well askt , 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 , &c. Wherefore not onely 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 my self 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 Honorable the Lord Brouncker , President 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 premisses , 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 , &c. 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 afore-said , 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 , then 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 Quadrato-quadratic ; 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 Nutshel ; ) I say , Weaker , for what is here said ; and Dearer , for what shall be said hereafter in the sixteenth Instance of Masts , Diamonds , &c. And on the other hand , if the Timbers were Quadratoquadratic , then 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 Engin , 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 , ought to have 16 times as much Timber in it , as the Cask which holds onely 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 , &c. they are usually the weaker compar'd 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 : ) Lastly , let the strength of the Timber be also given . Now the Questions are , to find the Scantlings of the Girders , Gise , &c. 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 Boards , but 1600. Now if the same Board were planck 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 planck 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 contein but half the stuff , which the 4½ inch-Planck first mentioned did contein . 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 onely to intimate the method of inquiring into these matters so useful in the world . The Third Instance ▪ In the Oars of a Boat , &c. 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 Skuller with his Skulls and Bench , &c. 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 Skuller 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 stroaks 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 , &c. supposing still the same number of stroaks , 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 heavs and sets , suppose 20 degrees of the Quadrant in it ; then , for asmuch as the Boats way will be encreased 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 burthen on his back ; then with half the said burthen he shall travel 7 ; and with double but three miles and a half . Again , suppose a Horse with 200 pound burthen can endure to travel 10 hours per diem ; then with half the same burthen 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 , then 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 Gun-powder 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 afore-mentioned . 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 , then 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 contein 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 abovemention'd 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 Musquets 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. Iames's 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 Battel 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 Doctrin 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 smelt one Perch or Rod ( whereof 320 make a mile , ) then about 8000 Acres of Land , whereon such sented 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 smelt 64 miles : And 72000 Acres of the like Land , or a parcel of such Land about 11 miles square , may be smelt as many leagues , or near 200 miles . And this Consideration I pitch upon , as one of the grounds whereupon I would build a Doctrin 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 , &c. 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 Doctrin ( 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 Center of the Pendulum , and the Center 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 Small-pox for the most part : And for that those of 26. are scarce come to the Gout , Stone , Dropsie , Palsies , Lethargies , Apoplexies , and other Infirmities of Old Age. Now whether these be sufficient reasons , is not the present Enquiry ; but taking the afore-mentioned Assertion to be true ; I say , that the Roots of every number of Mens Ages under 16 ( whose Root is 4 ) compared with the said number 4 , doth shew the proportion of the likelyhood of such mens reaching 70 years of Age. As for example ; 'T is 4 times more likely , that one of 16 years old should live to 70 , then a new-born Babe . 'T is three times more likely , that one of 9 years old should attain the said age of 70 , than the said Infant . Moreover , 't is 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 , 't is 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 accompts 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 Indexes for several times and places , would make an useful Scale of Salubrity for those places ; and a better Judg of Ayres 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 Musick . TAke a Musical String , one end thereof being fastned ; 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 8 th or Diapason above it self , 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 Mettal , so as able Artists can cast Bells in Tones assigned . The Thirteenth Instance , of Fire and Spirits . LEt a Cylindrical Flat-bottom Vessel be filled with Water , and let it be tried , in what time one Lamp or Candle would make the water boyl 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 Fire-works great Fires are more profitable than small ; as in Brewers 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 contein 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 shew 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 't is certain , that as the Spirit riseth double upwards , so also it emitteth or rarifieth it self 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 stroaks 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 stroak ( which stroaks 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 above-mentioned . 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 above-mentioned 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 Stroaks made within the Time of the waters sinking every such equal Space ; for therein the above-mentioned 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 , &c. 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 , &c. Wherefore to stop Leaks afore , the Ship must stop its motion , lye by , or bear up to go with the Wind and Sea , &c. 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 Doctrin , 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 Iron-work 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 heighth , and be worth 40 s ; then a Mast of 20 inches through , and double length also , shall not onely cost eight times as much , according to the Octuple quantity of Timber it conteins , 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 onely 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 Burthen to know her worth by the Tun , with the Number and Size of her Ordnance , &c. The Seventeenth Instance , In Mill-Dams , Sea-Bancks , and Bulwarks of Fortresses . SUppose any Wall , Dam , or Banck , 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 Damm 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 , then 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 Wooll , &c. SUppose some Cylindrical or other parallell'd sided Vessel , fill'd with Wool , or Down , or Feathers , or other Elastic Materials ; let the same be covered with a moveable 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 Thred 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 condens'd , 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 , &c. So as in a drawn Bow , let the Arrow be divided into quotcunque partes , 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 equalize the whole . The like also happens in the Traction of Muscles upon two Bones with a turning Joynt 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 subjoyn a short Appendix of Elasticity it self , 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 streight Lath , being by force bent circularly , doth upon the removal of that Force , return to be streight again by its Elasticity ; and a Circular Hoop being forced streight , 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 befal them . 6. I suppose in every Atome 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 Byas . The Heavens also visibly have their Poles , and must have a Center of Gravity or Magnitude , or some other Central and predominant Point . 7. I suppose every Atome 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 Iovis about Iupiter , &c. 8. I suppose , that the Byas of one Atome may have a tendency towards the Byas 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 Center of the Earth , and the other of Verticity , by which it tendeth towards the Earths-Poles , and whereby Magnets joyn to each other by their Opposite Poles . 10. All Atoms by their Motion of Verticity or Polarity , would draw themselves , like Magnets , into a streight 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 , curb them into a Triangle , whereof the Two Axes of Two Atoms are two sides , and the distance between the Byass of each making the third side : Wherefore I call the Polar Motion above-mentioned , 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 ballance 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 above-named 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 onely 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 Atome , 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 .