# A Treatise of the Elements of the Algebraical Art

John Kersey wrote A Treatise of the Elements of the Algebraical Art which was first published in 1673. Written in English rather than Latin, the book was clearly intended for a wide readership. We give below extracts from the first Chapter of Book I, to give the reader a feel for the style of Kersey's book which provides a wonderful view of 17th century algebra when the subject was moving towards the algebra we are familiar with today. We have modernised the English spelling, but tried to maintain the old English style.

Let us clarify the word Cossic which Kersey sometimes uses. A 'thing' in Latin is Causa, in Italian is Cosa and in German is Coss, and the 'thing' became the word for the 'unknown'. Those trying to find the 'unknown' were known as Cossists in early times and algebra was often called the cossic art. Let us also note that, of course, Renates des Cartes is René Descartes.

A Treatise of the Elements of the Algebraical Art

by

John Kersey
Book I   Chapter I.

Concerning the Nature, Scope, and Kinds of Algebra: The Construction of Cossic Quantities, or Powers; with the manner of expressing them by Alphabetical Letters: The signification of Characters used in the First Book.

I. The Mathematical Arts or Sciences are exercised about Quantity, which is comprised under Numbers, Lines, Superficies, and Solids: These if they be considered abstractly, and separate from all kind of Matter, are the proper Objects of Arithmetic and Geometry, which are called Pure Mathematics.

II. The Method which Mathematicians are wont to use in searching out Truth about Quantity, is twofold; viz. 1. Synthetical, or by way of Composition: 2. Analytical, or by way of Resolution.

III. Mathematical Composition, or the Synthetical Method, argues altogether with known Quantities to search out unknown, and then demonstrates that the Quantity found out will satisfy the Proposition.

IV. Mathematical Resolution, or the Analytical Art, commonly called Algebra, is that way of Reasoning which assumes or takes the Quantity sought, as if it were known or granted; and then with the help of one or more Quantities given or known, proceeds by Consequences, until at length the Quantity first only assumed or feigned to be known, is found equal to some Quantity or Quantities certainly known, and is likewise known.

V. The Scope, Drift or Office of the Analytic or Algebraic Art, is to search out three kinds of Truths, viz.

1. Theorems; which are nothing else but Declarations, or Affirmations of certain Properties, Proportions, or Equalities, justly inferred from some Supposition or Concessions about Quantity: Which Theorems are to be reserved store, as ready helps to find out new, and to confirm old Truths. This kind of Resolution when it rests in a bare Invention of Truth, is called Contemplative, or Notional.

2. Canons, or infallible Rules, to direct how to solve knotty Questions, by the help of Quantities given or known; this kind of Resolution is called Problematical.

3. Demonstrations, or evident and indubitable Proofs, to manifest the Truth of such Theorems and Canons as are Analytically found out.

VI. Algebra is by late Writers divided into two kinds; to wit, Numeral or Literal (or Specious).

VII. Numeral Algebra is so called, because in this Method of resolving a Question, the Quantity sought or unknown is solely designed or represented by some Alphabetical Letter, or other Character taken at Pleasure, but all the Quantities given are expressed by Numbers.

VIII. Literal or Specious Algebra is so called, because in this Method of resolving a Question, as well the given or known Quantities, as the unknown are all severally expressed or represented by Alphabetical Letters. Whence it comes to pass, that at the end of the Resolution of a Question, every Quantity appearing distinct under the same Letter or Form by which it was first expressed, a Canon is discovered to direct how the Question proposed may be solved, not only by the Quantities first given, but by any other whatsoever that are capable of solving the Question. In this Respect therefore Literal Algebra far excels the Numeral; for this latter serves only to solve Arithmetical Questions, and produces not a Canon without much difficulty, in regard the Numbers first given, by reiterated Multiplications, Divisions, and other Arithmetical Operations, will for the most part be so confounded and interwoven, that their Footsteps can hardly be traced out: But Literal or Specious Algebra is applicable to the solving of Geometrical Problems, as well as Arithmetical.

IX. The Doctrine of Algebra is principally grounded upon Knowledge of certain Quantities called by some Authors Cossic Quantities, by others, Powers; the Construction whereof is explained in six Sections following.

X. Numbers are said to be in Geometrical Proportion continued, when as the first is to the second, so is the second to the third, and so is the third to the fourth, etc. As for Example, these Numbers 1, 2, 4, 8, 16, 32, etc are Continual Proportionals; for; as the first Term 1, is the half of the second Term 2; so is the second Term 2, the half of the third Term 4; and so is 4 the half of 8, etc. Likewise these Numbers, 3, 9, 29, 81, 243, etc are in Geometrical Proportion continued; For as the first Term 3 is a third part of the second Term 9, so is the second Term 9 a third part of the third Term 27; and so is 27 one third of 81, etc. Also these Numbers are continual Proportionals, to wit, $1, \large\frac{1}{2}\normalsize , \large\frac{1}{4}\normalsize , \large\frac{1}{8}\normalsize$, etc. for as the first Term is 1, is the double of the second Term; so is $\large\frac{1}{2}\normalsize$ the double of $\large\frac{1}{4}\normalsize$, and $\large\frac{1}{4}\normalsize$ the double of $\large\frac{1}{8}\normalsize$, etc.

XI. In any series or rank of Numbers proceeding from Unity in a continued Geometrical Proportion, whether ascending or descending, all the Numbers or Terms except the first, which is supposed to be 1, (to wit, Unity,) are called Cossic Numbers, or Powers; viz. the second Term or Proportional is called the Root or first Power, the third Proportional is called the Square, or second Power, the fourth Proportional is called the Cube, or third Power; the fifth Proportional is called the Biquadrate; or fourth Power, the firth Proportional, the fifth Power, etc. As for Example, in this rank of Continual Proportionals, 1, 2, 4, 8, 16, 32, etc. the second Term 2 is the Root; the third Term 4 is the second Power, or the Square of the Root 2; the fourth Term 8 is the third Power, or the cube of the Root 2; the fifth Term 16 is the Biquadrate or fourth Power of the same Root 2 etc.

In like manner in this rank of Continual Proportionals descending from 1, to wit, $1, \large\frac{1}{2}\normalsize , \large\frac{1}{4}\normalsize , \large\frac{1}{8}\normalsize , \large\frac{1}{16}\normalsize$, etc the second Term $\large\frac{1}{2}\normalsize$ is the Root, the third Term $\large\frac{1}{4}\normalsize$ is the second Power, the fourth Term $\large\frac{1}{8}\normalsize$ is the third Power etc. The like is to be understood of any other Rank of Numbers in a continued Geometrical Proportion, whose first Term or Proportional is Unity.

XII. From the two last preceding Sections (which are grounded upon 10. Prop. 8 Elem. Euclid.) it is evident that any number whatsoever being proposed for a Root, the second Power, or the Square, is produced by the Multiplication of the Root by itself; the third Power, or the Cube, is produced by the Multiplication of the second Power by the Root; the fourth Power is produced by the Multiplication of the third Power by the Root, etc.

As for Example, if 2 be given for the Root, this 2 multiplied by itself, produces 4 for the second Power, to wit, the Square of the Root 2: Again, 4 the second Power being multiplied by the Root 2 gives 8 the third Power, or the Cube; which third Power multiplied by the Root 2, produces the fourth Power 16, etc.
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XVI. In searching out the Solution of a Question by the Algebraic Art, the Number or line sought is usually called a Root, which so long as it remains unknown cannot be really expressed, and therefore it must be designated or represented by some Symbol of Character, at the will of the Artist; also the Powers which may be imagined to proceed from the said Root in such manner as has before been declared are likewise to be represented by Symbols or Characters; concerning which there is much diversity among Algebraical Writers, every one pleasing his Fancy in the choice of Characters: But in this Matter I shall imitate Mr Thomas Harriot in his Ars Analytica, and Renates des Cartes in his Geometry, but chiefly the former; whose Method of expressing Quantities by Alphabetical Letters, I conceive to be the plainest for Learners, etc.

To design or represent the Root sought, whether it be a number or a Line in a Question proposed, we may assume any Letter of the Alphabet, as for $a, b$, or $c$, etc. but for the better distinguishing of known Quantities from unknown, some Analysts are wont to assume one of the five Vowels, as $a$, or $e$, etc. to represent the Quantity sought; and Consonants, as $b, c, d$, etc. to represent Quantities known or given: Now if the Letter $a$ be assumed to represent the Root sought, then (according to Mr Harriot) the second Power, or the Square raised from that Root, may be represented by aa; the third Power, or the Cube, by $aaa$; the fourth Power by $aaaa$; the fifth Power by $aaaaa$; and after the same manner any higher Power of the Root or Number $a$ may be represented: For so many Dimensions or Degrees as are in the Power, so many times the Letter which at first was assumed for the Root is to be repeated.

Or after the manner of Renates des Cartes, if the Letter $a$ be assumed to represent the Root, the Square may be designed thus $a^{2}$, the Cube thus, $a^{3}$, the fourth Power thus, $a^{4}$, the fifth Power thus, $a^{5}$. And so any other Power may be expressed by writing the Index or Exponent of the Power in a small Figure next after, and near the head of the Letter assumed to represent the Root.

After the same manner, known Quantities and their Powers may be represented by Consonants; as, $b$ may be put for any known number in a Question, and then its Square may be signified by $bb$, the Cube by $bbb$, the fourth Power by $bbbb$, the fifth Power by $bbbbb$, the sixth by $bbbbbb$, and so forwards: Or the Square of the Root $b$ may be expressed thus, $b^{2}$, the Cube thus, $b^{3}$, the fourth Power thus, $b^{4}$, the fifth Power thus, $b^{5}$, the sixth Power thus, $b^{6}$, and do forward.

XVII. Numbers set before, that is, on the left hand of quantities expressed by Letters are called Numbers prefixt; but if no Number be prefixt to the Letter, then 1 or unity must be imagined to be prefixt: As in these quantities $a$, (or $1a$,) $2a$, $3a$, $\large\frac{1}{2}\normalsize a$, $\large\frac{2}{3}\normalsize a$, $5bbb$ (or $5b^{3}$) the Numbers prefixt are (as you see) 1, 2, 3, $\large\frac{1}{2}\normalsize , \large\frac{2}{3}\normalsize$, and 5, every one of which Numbers (and the like so prefixt) shews how often the Quantity represented by the Letter or Letters immediately following the Number is taken; so $a$ or $1a$ signifies some Number or Line once taken, also $2a$ represents the double, $\large\frac{1}{2}\normalsize a$ the half, and $\large\frac{2}{3}\normalsize a$ two third parts of the Number or Line represented by $a$. In like manner $5bbb$, or $5b^{3}$, signifies that the Cube of the Number or Line represented by $b$ is taken five times.

XVIII. All numbers expressed by Figures and Cyphers (as in vulgar Arithmetic) not having any Letter or Letters annexed to them, are for distinction sake called Absolute numbers; as these numbers, 5, 20, 105, $\large\frac{1}{2}\normalsize$, $\large\frac{2}{3}\normalsize$, and all others when they be not prefixt or annext to any Letter or Letters are called absolute Numbers.

XIX. All Algebraical Operations are performed in an Arithmetical manner, partly in the vulgar way by Numbers, and partly by Alphabetical Letters in all parts of Arithmetic, to wit, Addition, Subtraction, Multiplication, Division, and the Extraction of Roots: But since Letters cannot be disposed like Numbers to perform those Operations, some Characters must of necessity be used to signify such Operations. The Characters used in this first Book are explained in the following Sections.

Last Updated September 2023