To the Right Honourable and others, the Learned Members of the Royal Society, for the Advancement of the Sciences.  
PResenteth to your Consideration, your most humble servant Thomas Hobbes, a Confutation of a Theorem which hath a long time passed for Truth; to the great hindrance of Geometry, and also of Natural Philosophy, which thereon dependeth.  

The Theorem.  
The four sides of a Square being divided into any number of equal parts, for example into 10; and strait lines drawn through the opposite points, which will divide the Square into 100 lesser Squares; The received Opinion, and which Dr. Wallis commonly useth, is, that the root of those 100, namely 10, is the side of the whole Square.  

The Confutation.  
The Root 10 is a number of those Squares, whereof the whole containeth 100, whereof one Square is an Unity; therefore the Root 10, is 10 Squares: Therefore the Root of 100 Squares is 10 Squares, and not the side of any Square; because the side of a Square is not a Superficies, but a Line. For as the root of 100 Unities is 10 Unities, or of 100 Soldiers 10 Soldiers: so the root of 100 Squares is 10 of those Squares. Therefore the Theorem is false; and more false, when the root is augmented by multiplying it by other greater numbers.  

Hence it followeth, that no Proposition can either be demonstrated or confuted from this false Theorem. Upon which, and upon the Numeration of Infinites, is grounded all the Geometry which Dr. Wallis hath hitherto published.  
And your said servant humbly prayeth to have your Judgement hereupon: And that if you find it to be false, you would be pleased to correct the same; and not to suffer so necessary a Science as Geometry to be stifled, to save the Credit of a Professor.      

Three PAPERS Presented to the ROYAL SOCIETY Against Dr. WALLIS. Together with CONSIDERATIONS ON Dr. Wallis his ANSWER to them.  
By THO. HOBBES of Malmsbury.  
LONDON: Printed for the Author; and are to be had at the Green Dragon without Temple-bar. 1671.   

To the Right Honourable and others, the Learned Members of the Royal Society, for the Advancement of the Sciences.  
YOur most humble servant Thomas Hobbes presenteth, That the quantity of a Line calculated by extraction of Roots, is not to be truly found. And further presenteth to you the Invention of a Strait Line equal to the Arc of a Circle.  
A Square Root is a number which multiplied into itself produceth a number. And the number so produced is called a Square number. 
Definition.  For example: Because 10 multiplied into 10 makes 100; the Root is 10, and the Square number 100  

Consequent.  
In the natural row of Numbers, as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, etc. every one is the Square of some number in the same row. But Square numbers (beginning at 1) intermit first two numbers, then four, then six, etc. So that none of the intermitted numbers is a Square number, nor has any Square root.  

PROP. I.  
A Square root (speaking of quantity) is not a Line, such as Euclid defines, without Latitude, but a Rectangle.  
Suppose A B C D be the Square, and AB, BC, CD, DAMN be the sides; and every side divided into 10 equal parts, and Lines drawn through the opposite points of division; there will then be made 100 lesser Squares, which taken all together are equal to the Square ABCD. Therefore the whole Square is 100, whereof one Square is an Unit; therefore 10 Units, which is the Root, is ten of the lesser Squares, and consequently has Latitude; and therefore it cannot be the side of a Square, which according to Euclid is a Line without Latitude.  

Consequent.  
It follows hence, that whosoever taketh for a Principle, That a Side of a Square is a mere Line without Latitude, and That the Root of a Square is such a Line, (as Dr. Wallis continually does) demonstrates nothing.  
But if a Line be divided into what number of equal parts soever, so the Line have breadth allowed it, (as all Lines must, if they be drawn) and the length be to the breadth as Number to an Unite, the Side and the Root will be all of one length.    

PROP. II.  
Any Number given is produced by the greatest Root multiplied into itself, and into the remaining Fraction. Let the Number given be two hundred Squares, the greatest Root is 14 4/14 Squares. I say, that 200 is equal to the product of 14 into itself, together with 14 multiplied into 4/14. For 14 multiplied into itself, makes 196. And 14 into 4/14 makes 56/14, which is equal to 4. And 4 added to 196 maketh 200; as was to be proved.  
Or take any other Number 8, the greatest Root is 2; which multiplied into itself is 4, and the Remainder 4/2 multiplied into 2 is 4; and both together 8.   

PROP. III.  
But the same Square calculated Geometrically by the like parts, consisteth (by Eucl. 2.4.) of the same numeral great Square 196, and of the two Rectangles under the greatest side 14, and the Remainder of the side, or (which is all one) of one Rectangle under the greatest side, and double the Remainder of the side; and further of the Square of the less Segment; which all together make 200, and moreover 1/49 of those 200 Squares, as by the operation itself appeareth thus.  
The side of the greater Segment is 

14¼.  
14¼.   Which multiplied into itself, makes 200.  
The product of 14 the greatest segment, into the two Fractions 4/14, that is, into 4/14 (or into twice 2/14) is 56/14 (that is 4) and that 4 added to 196 makes 200.  
Lastly, the product of 2/14 into 2/14 or 1/7 into 1/7, is 1/49.  
And so the same Square calculated by Roots, is less by 1/49 of one of those two hundred Squares, then by the true and Geometrical Calculation; as was to be demonstrated.  

Consequent.  
It is hence manifest, That whosoever calculates the length of an Arc or other Line by the extraction of Roots, must necessarily make it shorter than the truth, unless the Square have a true Root.      

The Radius of a Circle is a Mean Proportional between the Arc of a Quadrant and two fifths of the same.  
DEscribe a Square ABCD, and in it a Quadrant DCA. In the side DC take DT two fifths of DC; and between DC and DT a Mean Proportional DR; and describe the Quadrantal Arcs RS, TU.  
I say, the Arc RS is equal to the straight line DC.  
For seeing the proportion of DC to DT is duplicate of the proportion of DC to DR, it will be also duplicate of the proportion of the Arc CA to the Arc RS; and likewise duplicate of the proportion of the Arc RS to the Arc TU.  
Suppose some other Arc less or greater than the Arc RS to be equal to DC, as for example rs: Then the proportion of the Arc rs to the straight line DT will be duplicate of the proportion of RS to TV, or DR to DT. Which is absurd; because Dr is by construction greater or less than DR.  
Therefore the Arc RS is equal to the side DC. Which was to be demonstrated.  

Corol.  
Hence it follows that DR is equal to two fifths of the Arc CA For RS, TV, DT being continually proportional; and the Arc TV being described by DT, the Arc RS will be described by a straight line equal to TU. But RS is described by the straight line DR. Therefore DR is equal to TV, that is, to two fifths of CA  
And your said servant most humbly prayeth you to consider (if the demonstration be true and evident) whether the way of objecting against it by Square Roots, used by Dr. Wallis; and whether all his Geometry, as being, built upon it, and upon his supposition of an Infinite Number, be not false.    

Considerations upon the Answer of Dr. Wallis to the Three Papers of Mr. Hobbes.  
DR. Wallis says, All that is affirmed, is but, If we SUPPOSE That, This will follow.  
But it seemeth to me, that if the Supposition be impossible, then that which follows will either be false, or at least undemonstrated.  
First, this Proposition being founded upon his Arithmetica Infinitorum, If there he affirm an absolute Infiniteness, he must here also be understood to affirm the same. But in his 39th Proposition he saith thus: Seeing that the number of terms increasing, the excess above sub-quadruple is perpetually diminished, so as at last it becomes less than any proportion that can be assigned; If it proceed in infinitum it must utterly vanish. And therefore if there be propounded an Infinite row of quantities in triplicate proportion of quantities Arithmetically proportional (that is, according to the row of Cubical numbers) beginning from a point or 0; that row shall be to a row of as many, equal to the greatest, as 1 to 4.  
It is therefore manifest that he affirms, That in an Infinite row of quantities the last is given; and he knows well enough that this is but a shift.  
Secondly, he says, That usually in Euclid and all after him, by Infinite is meant but, more than any assignable Finite, or the greatest possible.  
I am content it be so interpreted. But then from thence he must demonstrate those his conclusions, which he hath not yet done. And when he shall have done it, not only the Conclusions, but also the Demonstration will be the same with mine in Cap. 14. Art. 2, 3, etc. of my Book De Corpore. And so he steals what he once condemned. A fine quality.  
Thirdly, he says (by Euclides 10th Proposition, but he tells not of what Book) That a Line may be bisected, and the halves of it may again be bisected, and so onwards infinitely; and that upon such supposed Section Infinitely continued, the parts must be supposed Infinitely many.  
I deny that; for Euclid, if he says a Line may be divisible into parts perpetually divisible, he means, That all the divisions, and all the parts arising from those divisions, are perpetually Finite in number.  
Fourthly, he says, That there may be supposed a row of quantities infinitely many, and continually increasing, whereof the last is given.  
'Tis true, a man may say (if that be supposing) that white is black; but if Supposing be Thinking, he cannot suppose an Infinite row of quantities whereof the last is given. And if he say it, he can demonstrate nothing from it.  
Fifthly, He says (for one absurdity begets another) That a Superficies or Solid may be supposed so constituted, as to be Infinitely long, but Finitely great (the breadth continually decreasing in greater proportion than the length increaseth) and so as to have no centre of gravity. Such is Toricellio 's Solidum Hyperbolicum acutum, and others innumerable discovered by Dr. Wallis, Monsieur Fermat, and others. But to determine this, requires more of Geometry and Logic (whatsoever it do of the Latin Tongue) than Mr. Hobbes is master of.  
I do not remember this of Toricellio, and I doubt Dr Wallis does him wrong, and Monsieur Fermat too. For to understand this for sense, 'tis not required that a man should be a Geometrician or a Logician, but that he should be mad.  
In the next place he puts to me a Question as absurd as his Answers are to mine. Let him ask himself (saith he) if he be still of opinion, That there is no Argument in Natural Philosophy to prove that the World had a beginning: First, whether in case it had no beginning, there must not have passed an Infinite number of years before Mr. Hobbes was born. Secondly, whether at this time there have not passed more, that is, more than that Infinite number. Thirdly, whether, in that Infinite (or more than Infinite) number of years, there have not been a greater number of days and hours, and of which hitherto the last is given. Fourthly, whether, if this be an Absurdity, we have not then (contrary to what Mr. Hobbes would persuade us) an Argument in Nature to prove the World had a beginning.  
To this I answer, not willingly, but in service to the Truth, that by the same Argument he might as well prove that God had a beginning. Thus: in case he had not, there must have passed an Infinite length of time before Mr. Hobbes was born; but there hath passed at this day more than that Infinite length (by eighty four years). And this day, which is the last, is given. If this be an Absurdity, have we not then an Argument in Nature to prove that God had a beginning? Thus 'tis when men entangle themselves in a Dispute of that which they cannot comprehend. But perhaps he looks for a Solution of his Argument to prove that there is somewhat greater than Infinite; which I shall do so far, as to show it is not concluding. If from this day backwards to Eternity be more than Infinite, and from Mr. Hobbes his birth backwards to the same Eternity be Infinite, then take away from this day backwards to the time of Adam, which is more than from this day to Mr. Hobbes his birth, then that which remains backwards must be less than Infinite. All this arguing of Infinites is but the ambition of Schoolboys.  

To the latter part of the first Paper.  
There is no doubt, if we give what Proportion we will of the Radius to the Arc, but that the Arc upon that Arc will have the same Proportion. But that is nothing to my Demonstration. He knows it, and wrongs the Royal Society in presuming they cannot find the Impertinence of it.  
My proof is this; That if the Arc on TV, and the Arc RS, and the straight Line CD, be not equal, than the Arc on TV, the Arc on RS, and the Arc on CA, cannot be proportional. Which is manifest by supposing in DC a less than the said DC; but equal to RS, and another straight Line, less than RS, equal to the Arc on TV; and any body may examine it by himself.  
I have been asked by some that think themselves Logicians, Why I proceeded upon 2/5 rather than any other part of the Radius. The reason I had for it was, That long ago some Arabians had determined, That a straight Line whose square is equal to 10 squares of half the Radius, is equal to a quarter of the Perimiter; but their demonstrations are lost. From that Equality it follows, that the third proportional to the Quadrant and Radius, must be a mean proportional between the Radius and 2/5 of the same. But my answer to the Logicians was, That though I took any part of the Radius to proceed on, and lighted on the Truth by chance, the Truth itself would appear by the Absurdity arising from the denial of it. And this is it that Aristotle meant, where he distinguisheth between a Direct demonstration, and a demonstration leading to an absurdity. Hence it appears, that Dr. Wallis his objections to my Rosetum are invalid, as built upon Roots.   

To the second Paper.  
First, he says, That it concerns him no more than other men. Which is true. I meant it against the whole Herd of them who apply their Algebra to Geometry.  
Secondly, He says, That a bare Number cannot be the Side of a Square figure.  
I would know what he means by a Bare Number. Ten Lines may be the side of a Square figure. Is there any Number so bare, as by it we are not to conceive or consider any thing numbered? Or by ten Nothings understands he Bore 10? He struggles in vain, his Conscience puzzles him.  
Thirdly, He says, Ten Squares is the Root of 100 Square-squares. To which I answer, first, That there is no such Figure as a Square-square. Secondly, That it follows hence that a Root is a Superficies, for such is 10 Squares.  
Lastly, He says, That neither the Number 10, nor 10 Soldiers is the Root of 100 Soldiers; because 100 Soldiers is not the Product of 10 Soldiers into 10 Soldiers.  
That last I grant, because nothing but Numbers can be multiplied into one another. A Soldier cannot be multiplied by a Soldier. But no more can a Square-figure by a Square-figure, though a Square-number may. Again, If a Captain will place his hundred Men in a square Form, must not he take the Root of 100 to make a Rank or File? And are not those 10 Men?   

To the third Paper.  
He objects nothing here, but that, The Side of a Square is not a Superficies but a Line, and that a Square Root (speaking of quantity) is not a Line but a Rectangle, is a contradiction. The Reader is to judge of that.  
To his Scoffing I say no more, but that they may be retorted in the same words, and are therefore childish.  
And now I submit the whole to the Royal Society, with confidence that they will never engage themselves in the maintenance of these Unintelligible Doctrines of Dr. Wallis, that tend to the suppression of the Sciences which they endeavour to advance.