WHich that we may more plainly shew, especially as far as it relates to the Rules of Extraction, consider, 1. That the Root of the Square aa+2ab+bb is already known (for we assumed for the Root the Quantity a+b) so that now we are to see which way this Root is to be obtain'd out of that Square by Division. It will presently appear, that the first Note of the Root a, will come out of the first part of the Square aa, and the other part b must be obtain'd out of the remainder 2ab+bb; and so as there are 2 Notes of the Root, the Square must be distinguish'd as it were into 2 Classes, each of which gives a particular note of the Root. Then it is manifest, that the first Note of the Root (a) may be obtain'd out of the Square aa by a simple Extraction. Now it is e∣vident, if I would have by Division the other Note of the Root, the next following part of the remaining Classis must be divided by 2a, the double of the Quotient just now found, and that nothing should remain after this Division (for now we have the whole Root a+b) you must not only subtract the Product of the Divisor and this new Quotient, but also the Square of this new Quotient: Which is the Vulgar Method and Rule for the Extraction of Square Roots taught in common Arithmetick.
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 13, 2025.
Pages
Page 47
Likewise if you would extract the Root of the above-men∣tioned Cube, which we already know, having formed it from a+b, it is manifest, that the first Note of the Root a will come out of the first part of the Cube a{powerof3}, and the other b, must be obtain'd out of the remainder 3abb+bbb, and so, as there are two Notes of the Root, the Cube must be distinguish'd, as it were into two Classes, each of which will give a particular Note of the Root. Now it is manifest, the first Note of the Root a is obtained by simple Extraction of the Root out of the Cube aaa. It is moreover evident, if I would have by Division the other Note of the Root b, the next remaining part must be di∣vided by 3aa (the triple Square of the precedent Quotient, or thrice the precedent Quotient multiplied by it self) and, that nothing should remain after this division (for now we have the whole Cube Root a+b) you must not only subtract from the remaining Dividend the Product of the Divisor, and the new Quotient (3aab) but also the Product of the Square of the new Quotient, and thrice the precedent Quotient (3abb) and more∣over the Cube of that new Quotient b{powerof3}: Which is the Method of extracting Cube Roots in Vulgar Arithmetick.