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A44014
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Six lessons to the professors of the mathematiques one of geometry the other of astronomy, in the chaires set up by the noble and learned Sir Henry Savile in the University of Oxford.
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Hobbes, Thomas, 1588-1679.
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1656
(1656)
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Wing H2260; ESTC R22309
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102,047
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75
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SIX LESSONS To the PROFESSORS of the MATHEMATIQUES ONE OF GEOMETRY THE OTHER OF ASTRONOMY In the Chaires set up by the Noble and Learned Sir HENRY SAVILE in the University of Oxford LONDON Printed by J. M. for Andrew Crook at the Green-Dragon in Pauls Church-yard To the Right Honourable Henry Lord Pierrepont Viscount Newarke Earle of Kingstone and Marquis of Dorchester My most Noble Lord NOt knowing on my own part any cause of the favour your Lordship has been pleased to express towards me unless it be the Principles Method and Manners you have observed and approved in my Writings and seeing these have all been very much reprehended by men to whom the name of Publique Professors hath procured reputation in the University of Oxford I thought it would be a forfeiture of your Lordships good opinion not to justifie my self in publique also against them Which whether I have sufficiently performed or not in the six following Lessons addressed to the same Professors I humbly pray your Lordship to consider The volume it self is too small to be offered to you as a Present but to be brought before you as a Controversie it is perhaps the better for being short Of Arts some are demonstrable others indemonstrable and demonstrable are those the construction of the Subject whereof is in the power of the Artist himself who in his demonstration does no more but deduce the Consequences of his own operation The reason whereof is this that the Science of every Subject is derived from a praecognition of the Causes Generation and Construction of the same and consequently where the Causes are known there is place for Demonstration but not where the Causes are to seek for Geometry therefore is demonstrable for the Lines and Figures from which we reason are drawn and described by our selves and Civill Philosophy is 〈…〉 we make the Common-wealth our selves But because of Naturall Bodies we know not the Construction but seek it from the Effects there lyes no demonstration of what the Causes be we seek for but onely of what they may be And where there is place for Demonstration if the first Principles that is to say the Definitions contain not the Generation of the Subject there can be nothing demonstrated as it ought to be And this in the three first Definitions of Euclide sufficiently appeareth For seeing he maketh not nor could make any use of them in his Demonstrations they ought not to be numbered among the Principles of Geometry And Sextus Empiricis maketh use of them misunderstood yet so understood as the said Professors understand them to the overthrow of that so much renouned Evidence of Geometry In that part therefore of my Book where I treat of Geometry I thought it necessary in my Definitions to express those Motions by which Lines Superficies Solids and Figures were drawn and described little expecting that any Professor of Geometry should finde fault therewith but on the contrary supposing I might thereby not only avoid the Cavils of the Scepticks but also demonstrate divers Propositions which on other Principles are indemonstrable And truly if you shall finde those my Principles of Motion made good you shall find also that I have added something to that which was formerly extant in Geometry For first from the seventh Chapter of my Book de Corpere to the thirteenth I have rectified and explained the Principles of the Science id est I have done that business for which Doctor Wallis receives the wages In the seventh I have exhibited and demonstrated the proportion of the Parabola and Parabolasters to the Parallelograms of the same height and base which though some of the propositions were extant without their demonstration were never before demonstrated nor are by any other then this method demonstrable In the eighteenth as it is now in English I have demonstrated the for any thing I yet perceive Equation between the crooked line of a Parabola or any Parabolaster and a straight line In the twenty-third I have exhibited the Center of Gravity of any Sector of a Sphere Lastly the twenty-fourth which is of the nature of Refractiand Reflexion is almost all new But your Lordship will ask me what I have done in the twentieth about the Quadrature of the Circle Truely my Lord not much more then before I have let stand there that which I did before condemn not that I think it exact but partly because the Division of Angles may be more exactly performed by it then by any organicall way whatsoever and I have attempted the same by another Method which seemeth to me very naturall but of calculation difficult and slippery I call them only Aggressions retaining nevertheless the formall manner of Assertion used in Demonstration For I dare not use such a doubtfull word as Videtur because the Professors are presently ready to oppose me with a Videtur quod non Nor am I willing to leave those Aggressions out but rather to try if it may be made pass for lawfull in spight of them that seek honour not from their own performances but from other mens failings amongst many difficult undertakings carryed through at once to leave one and the greatest for a time behind and partly because the method is such as may hereafter give further light to the finding out of the exact truth But the Principles of the Professors that reprehend these of mine are some of them so void of sense that a man at the first hearing whether Geometrician or not Geometrician must abhor them As for example 1. That two equall Proportions are not double to one of the same Proportions 2. That a Proportion is double triple c. of a Number but not of a Proportion 3. That the same Body without adding to it or taking from it is sometimes Greater and sometimes less 4. That a Quantity may grow less and less Eternally so as at last to be equall to another Quantity or which is all one that there is a Last in Eternity 5. That the nature of an Angle consisteth in that which lyes between the lines that comprehend the Angle in the very point of their concourse that is to say An Angle is the Superficies which lyes between the two Points which touch or as they understand a Point the Superficies that lyes between the two Nothings which touch 6. That the Quoâ—ient is the Proportion of the Division to the Dividend Upon these and some such other Principles is grounded all that Doctor Wallis has said not onely in his Elenchus of my Geometry but also in his Treatises of the Angle of Contact and in his Arithmetica Infinitorum which two last I have Fere in two or three leaves wholly and cleerly confuted And I verily believe that since the beginning of the world there has not been nor ever shall be so much absurdity written in Geometry as is to be found in those books of his with which there is so much presumption joyned that an 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉
of the like conjunction cannot be expected in less then a Platonick year The cause whereof I imagine to be this that he mistook the study of Symboles for the study of Geometry and thought Symbolicall writing to be a new kinde of Method and other mens Demonstrations set down in Symboles new Demonstrations The way of Analysis by Squares Cubes c. is very antient and usefull for the finding out whatsoever is contained in the nature and generation of rectangled Plains which also may be found without it and was at the highest in Vieta but I never saw any thing added thereby to the Science of Geometry as being a way wherein men go round from the Equality of rectangled Plains to the Equality of Proportion and thence again to the Equality of rectangled Plains wherein the Symboles serve only to make men go faster about as greater Winde to a Winde-mill It is in Sciences as in Plants Growth and Branching is but the Generation of the Root continued nor is the Invention of Theoremes any thing else but the knowledge of the Construction of the Subject prosecuted The unsoundness of the Branches are no prejudice to the Roots nor the Faults of Theoremes to the Principles And active Principles will correct false Theoremes if the Reasoning be good but no Logique in the world is good enough to draw evidence out of false or unactive Principles But I detain your Lordship too long For all this will be much more manifest in the following Discourses wherein I have not onely explained and rectified many of the most important Principles of Geometry but also by the examples of those errors which have been committed by my Reprehenders made manifest the evil Consequence of the Principles they now proceed on So that it is not only my own Defence that I here bring before you but also a positive doctrine concerning the true Grounds or rather Atomes of Geometry which I dare only say are very singular but whether they be very good or not I submit to your Lordships judgement And seeing you have been pleased to bestow so much time with great success in the reading of what has been written by other men in all kindes of Learning I humbly pray your Lordship to bestow also a little time upon the reading of these few and short Lessons and if your Lordship finde them agreeable to your Reason and Judgement let me notwithstanding the clamour of my Adversaries be continued in your good opinion and still retain the honour of being My most Noble Lord London Iune 10. 1656. Your Lordships most Humble and Obliged Servant THOMAS HOBBES ERRATA PAge 2. l. 11. for Art 〈◊〉 act l. 12. for 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 p. 6. 〈◊〉 23. for Mr. 〈◊〉 Sir p. 21. l. 22. for Prâ—position r. Proportion p. 55. l. 20. for Senctoribus r. Senatâ—ribus p. 56. l. 8. for Serberius r. Sorberius LESSONS OF THE PRINCIPLES OF Geometry c. To the egregious Professors of the Mathematicks one of Geometry the other of Astronomy in the Chaâ—rs set up by the Noble and Learned Sir Henry Savile in the University of Oxford LESSON I. I Suppose most egregious Professors you know already that by Geomerry though the word import no more but the measuring of Land is understood no less the measuring of all other Quantity then that of Bodies And though the Definition of Geometry serve not for proof nor enter into any Geometricall Demonstâ—ation yet for understanding of the Principles of the Science and for a Rule to judge by who is a Geometrician and who is not I hold it necessary to begin therewith Geometry is the Science of determining the quantity of any thing not measured by comparing it with some other Quantity or Quantities measured Which Science therefore whosoever shall go about to teach must first be able to tell his Disciple what Measuring or Dimension is by what each several kind of Quantity is Measured what Quantity is what are the several kinds thereof Therefore as they who handle any one part of Geometry determine by Defiaition the signifie cation of every word whiÌ„ch they make the Subject or Praedicate of any Theoreme they undertake to demonstrate so must he which intendeth to write a whole body of Geometry Define and Determine the meaning of whatsoever word belongeth to the whole Science The design of Euclid was to demonstrate the Properties of the five regular bodies mentioned by Plato in which Demonstrations there was no need to alledge for Argument the Definition of Quantity which it may be was the cause he hath not any where Defined it but done what he undertook without it And though having perpetually occasion to speak of measure he hath not Defined Measure yet instead thereof he hath in the beginning of his first Element assumed an Axiome which serveth his turn sufficiently as to the measure of lines which is the eighth Axiome That those things which lye upon one another all the way called by him 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 are equall Which Axiome is nothing else but a description of the Art of Measuring Length and Superficies For this 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 can have no place in solid bodies unless two bodies could at the same time be in one place But amongst the Principles of Geometry universall the Definitions are necessary both of Quantity and Dimension Quantity is that which is signified by what we answer to him that asketh How much any thing is and thereby determine the magnitude thereof For magnitude being a word indefinite if a man ask of a thing Quantum est that is How much it is we do not satisfie him by saying it is magnitude or quantity but by saying it is Tantum so much And they that first called it in Greek 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and in Latine Quaâ—tity might more properly have called it in Latine Tantity and in Greek 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and we if we allowed our selves the Eloquence of the Greeks and Latines should call it the So-muchness There is therefore to every thing concerning which a man may ask without absurdity how much it is a certain Quantity belonging determining the magnitude to be so much Also wheresoever there is more and less there is one kinde of Quantity or other And first there is the Quantity of Bodies and that of three kindes Length which is by one way of Measuring Superficies made of the complication of two Lengths or the Measure taken two wayes and Solid which is the complication of three lengths or of the measure taken three wayes for breadth or thickness are but other Lengths And the Science of Geometry so far forth as it contemplateth Bodies onely is no more but by Measuring the length of one or more lines and by the position of others known in one and the same Figure to Determine by ratiocination how much is the Superficies and by Measuring Length Breadth and Thickness to determine the Quantity of the whole