FRom what we have said you have also the Reason of ano∣ther rule in Arithmetick which teaches how to approach continually nearer and nearer to the Square and Cube Roots of numbers that are not exact Squares and Cubes; viz. by adding to the given Number perpetually new Classes and Cyphers or o's, two at a time, to the Square, and three to the Cube, and so continue on the operation as before; which will add Deci∣mal Parts to the Integrals before found; and the next opera∣tion (if you add a second Classe of Cyphers) will exhibit Cen∣tesimal Parts, and so on ad Infinitum. For Example, If I would have the Square Root of 2 pretty near, I can assign no nearer whole Number than 1. But by adding a new Classe of 2 Cy∣phers, i. e. multiplying the given Number by 100 (whereby the Root is multiplied by 10) you'l have 14, nearly the Root of 200, that is, 14/10 or 1 4/10 much nearer the Root of 2 than the former; and thus you may always come nearer and nearer ad Infinitum, but never to an exact Root. For if you could have
Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.
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- Title
- Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.
- Author
- Sturm, Johann Christophorus, 1635-1703.
- Publication
- London :: Printed for Robert Knaplock and Dan. Midwinter and Tho. Leigh,
- 1700.
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- Subject terms
- Mathematics -- Early works to 1800.
- Geometry -- Early works to 1800.
- Algebra -- Early works to 1800.
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http://name.umdl.umich.edu/A61912.0001.001
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"Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S." In the digital collection Early English Books Online 2. https://name.umdl.umich.edu/A61912.0001.001. University of Michigan Library Digital Collections. Accessed May 18, 2025.
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Page 48
the exact Root of 2, or 3, or 5, &c. in any Fraction what∣soever, that Fraction must be of such sort, that its Numerator and Denominator being squared, the Fraction thence arising must exactly equal 2 or 3, or 5, &c. that is, its Numera∣tor must be exactly double, or trible, or Quadruple, &c. of of the Denominator; which can never be, because both are Squares, and in a Series of Squares no such thing can happen. Hence you have these