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A44011 Seven philosophical problems and two propositions of geometry by Thomas Hobbes of Malmesbury ; with an apology for himself and his writings. Hobbes, Thomas, 1588-1679. 1682 (1682) Wing H2259; ESTC R28663 37,975 99

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its own sides so that the Vnit which was before sometimes a Line sometimes a Square is now a Cube B. It can be no otherwise when you so apply Arithmetick to Geometry as to mumber the Lines of a Plain or the Plains of a Cube A. In the next place I find that the Cube of DY is equal to 45 want the Square Root of 1622. What is that 45 Lines or Squares or Cubes B. Cubes Cubes of DV A. Then if you add to 45 Cubes of DV the Square Root of 1682 the sum will be 45 Cubes of DV And if you add to the Cube of DY the same Root of 1682 the sum will be the Cube of DY plus the Square Root of 1682. And these two sums must be equal B. They must so A. But the Square Root of 1682 being a Line adds nothing to a Cube therefore the Cube alone of DY which he says is equal almost to 4. Cubes of DV is equal to 45 Cubes of the same DV B. All these impossibilities do necessarily follow the confounding of Arithmetick and Geometry A. I pray you let me see the Operation by which the Cube of DY that is the Cube of 3 want the Root of 2 is found equal to 45 want the Square Root of 1682. A detection of the absurd use of Arithmetick as it is now applyed to Geometry B. Here it is 3 r. 2. 3 r. 2. r. 18 ✚ 2 9 r. 18 9 r. 72 ✚ 2. 3 r. 2. r. 162 ✚ 12 r. 8. 27 r. 648 ✚ 6 27 r. 658 r. 162 ✚ 18 r. 8. A. Why for two Roots of 18 do you put the Root of 72. B. Because 2 Roots of 18 is equal to one Root of 4 times 18 which is 72. A. Next we have That the Root of 2 Multiplyed into 2 makes the Root of 8. How is that true B. Does it not make 2 Roots of 2 And is not BR the Root of 2 and 2 BR equal to the Diagonal And is not the diagonal the root of a square equal to 8 squares of DV A. 'T is true But here the root of 8 is put for the Cube of the root of 2. Can a line be equal to a Cube B. No. But here we are in Arithmetick again and 8 is a Cubique number A. How does the root of 2 multiplyed into the root of 72 make 12 B. Because it makes the root of 2 times 72 that is to say the root of 144 which is 12. A. How does 9 roots of 2 make the root of 162 B. Because it makes the root of 2 squares of 9 that is the root of 162. A. How does 3 roots of 72 make the root of 648 B. Because it makes the root of 9 times 72 that is of 648. A. For the total Sum I see 27 and 18 which make 45. Therefore the root of 648 together with the root of 162● and of 8 which are to be deducted from 45 ought to be equal to the root of 1682. B. So they are For 648 multiplyed by 162 makes 104976 of which the double root is 648 and 648 and 162 added together make 810. Therefore the root of 948 added to the root of 162 makes the root of 1459 Again 1458 into 8 is 11664. The double root whereof is 216. The Sum of 1458 and 8 added together is 1466. The Sum of 1466 and 216 is 1682 and the root the root of 1682. A. I see the Calculation in numbers is right though false in lines The reason whereof can be no other then some difference between multiplying numbers into lines or plains and multiplying lines into the same lines or plains B. The difference is manifest For when you multiply a number into lines the product is lines as the number 2 multiplyed into 3 lines is no more then 3 lines 2 times told But if you multiply lines into lines you make plains and if you multiply lines into plains you make solid bodies In Geometry there are but three dimensions Length Superficies and Body In Arithmetick there is but one and that is Number or Length which you will And though there be some Numbers called Plain others Solid others Plano-solid others Square others Cubique others Square-square others Quadrato-cubique others Cubi-cubique c. yet are all these but one dimension namely Number or a file of things Numbered A. But seeing this way of Calculation by Numbers is so apparently false what is the reason this Calculation came so near the truth B. It is because in Arithmetick Units are not Nothings and therefore have breadth And therefore many Lines set together make a superficies though their breadth be insensible And the greater the number is into which you divide your Line the less sensible will be your errour A. Archimedes to find a streight Line equal to the circumferrence of a Circle used this may of extracting Roots And 't is the way also by which the Table of Sines Secants aud Tangents have been calculated Are they all Cut B. As for Archimedes there is no man that does more admire him then I do But there is no man that cannot Err. His reasoning is good But he ads all other Geometricans before and after him have had two Principles that cross one another when they are applyed to one and the same Science One is that a Point is no part of a Line which is true in Geometry where a part of a Line when it is called a Point is not reckoned another is that a Unit is part of a Number which is also true but when they reckon by Arethmetick in Geometry there a Unit is somtimes part of a Line sometimes a part of a Square and sometimes part of a Cube As for the Table of Sines Secants and Tangents I am not the first that find fault with them Yet I deny not but they are true enough for the reckoning of Acres in a Map of Land A. What a deal of Labour has been lost by them that being Professors of Geometry have read nothing else to their Auditors but such stuff as this you have here seen And some of them have written great Books of it in strange characters such as in troublesome times a man would suspect to be a Cypher B. I think you have seen enough to satisfie you that what I have written heretofore concerning the Quadrature of the Circle and of other Figures made in imitation of the Parabola has not been yet confuted A. I see you have wrested out of the hands of our Antagonists this weapon of Algebra so as they can never make use of it again Which I consider as a thing of much more consequence to the science of Geometry then either of the Duplication of the Cube or the finding of two mean Proportionals or the Quadrature of a Circle or all these Problems put together FINIS Books Written by this Author and Printed for William Crooke 1 DE Mirabilibus Pecci in Quarto Latin in Octavo in English and Lantin 2 Three Papers to the Royal Society against Dr. Wallis 3 Lumathematica 4 Prima partis Doctrinae Wallisianae de Motu Censura Brevis 5 Resetum Geometricum sive propositiones aliquot frustra antehac tentatae 6 Principia Poblemata aliquot Geometrica ante desperata nunc breviter explicata demonstrata 7 Quadratura Circuli Cubatio Sphaere duplicatio Cubi Breviter demonstrata 8 Consideration on his Loyalty Reputation Religion and Manners by himself 9 De Principio Ratione Geometri 20 The Travels of Vlises Translated from Homer 11 Epistol ad Dr. Wood. 12 The Translation of all Homers Works into English 13 The Epitome of the Civil Wars of Enland from 1640 to 1660. 14 Aristotles Rhetorick Translated into English by him with his own Rhetorick to it 15 A Dialogue betwixt a Student in the Common-Laws of England and a Philosopher in which is set forth the Errors in some Practise 16 A Narration of Heresie and the Punnishment thereof 17 Ten Dialogues of Natural Philosophy 18 A Poem in Latin of his Life 19 idem the same in English 20 His Life written in Latin part by himself and the rest by Dr. R. B. wherein is contained the most material parts of his Life 21 Seven Philosophical Problems and two Propositions of Geometry With an Apology for Himself and his Writings Dedicated to the King in the year 1662.