95
rectanglej, &c. to which they refer. From the first propositionof the second book of Kudid, the nine following may be easilyderived in this manner, and they may he considered as properexamples of this most obvious application of algebra togeometry.
if there be certain positions either supposed or to be inferredin a theorem, we must find, according to the preceding observa-tions, the connection between these positions and such relations otmagnitude, as can he expressed and reasoned upon by algebra.The algebraical demonstrations, of the 12ih and 1.5th propositionsof the 2d book of Euclid, require only the 47th ot the I. El.The 33th and 3Glh of the 3d hook require only the 3. III. El.and 47. I. El.
From a few simple geometrical principles alone, a number ofconclusions, with regard to figures, may be deduced by algebra;and to this in a great measure is owing the extensive use of thisscience in geometry. If other more remote geometrical princi-ples be occasionally introduced, the algebraical calculations maybe much abridged, The same is to be observed in the solutionof problems ; but such in general are less obvious, and more pro-perly belong to the strict geometrical method.
(37.) Solution of Ph. oei.ems.
Upon the same principles are geometrical problems to be re-solved. The problem is supposed to be constructed, and pro-per algebraical notations of the known and unknown magnitudesare to be sought for,, by means of which their connections maybe expressed by equations. It may first be remarked, as wasdone in the case of theorems, that in those problems which relateto the division of the line and the proportions ot its parts, the ex-pression of the quantities, and the stating their relations by equa-tions, are so easy as not to require auv particular directions.But when various positions of geometrical figures and their pro-perties are introduced, the solution requires more attention andskill. No general rules can be given on this subject, but the fol-lowing observations may be of use:
1. 'The construction of the problem being supposed, it is oftenfarther necessary to produce some of the lines till they meet; todraw new lines joining remarkable points; to draw-'lines fromsuch points perpendicular or parallel to other lines, and suchother operations as seem conducive to the finding of equations;and for this purpose, those especially are to be employedwhich divide the scheme into triangles that are given, right-angled or similar.
It is often convenient to denote by letters, not the quan-tities particularly sought, but' some others from which they caneasily be deduced. The same may be observed of givenquantities.
3. The proper notation being made, the necessary equationsare to he derived by the use of the most simple geometricalprinciples; sue!) as the addition and subtraction of lines or ofsquares, the proportionality of lines, particularly of the sides ofsimilar triangles, fie. ,
4. There must be as many independent equations as there areunknown quantities assumed in the investigation, and from thesea filial equation may he inferred by the rules of Part 1.
If the final equation from the problem be resolved, the rootsmay oiten be exhibited geometrically ; but the geometrical con-rtruction of problems may he effected also without resolving theequation, and even without deducing a final equation.
If the final equation be simple or quadratic, the roots beingobtained by the common rules, may be geometrically exhibitedby the finding of proportionals, and the "addition or subtractionof squares.
By inserting numbers for the known quantities, a numeral ex-pression of the quantities sought will be obtained by resolvingthe equation. But in order to determine some particulars of theproblem, besides finding the unknown quantities of the equation,it may lie farther necessary to make a simple construction '; or, ifit be required that every thing be expressed in numbers, to sub-stitute a new calculation in place of that construction. bee Con-struction of Equations .
Psop. I. To divide a given straight line AB into two parts, so
that the rectangle contained by the whole line and one of the
parts may be equal to the square of the other part.
9
c A C B
Let C be the point of division, and let AB=fl, AC—r, andthen CB — a— x. From the problem a"— ax=-x- ; and this equa-tion being resolved gives a—T:
/ ‘A «
\ /a '-1 -•
4 2
/
The quantity -is the hypolhenuse of a right-angled
4
a
triangle, of which the two sides are a and —, and is therefore
t?
a
easily found; — being taken from this line, gives x— AC, which
o
is the proper solution. But if a line Ac be taken on the oppositeside of A, and equal to the above-mentioned hypolhenuse, togc-o / a‘ a
ther with — it will represent the negative root f ———,n ‘ 4 2
and will give another solution ; for in this case also ACx Be =Ac-. But c is without the line AB; and therefore, if i( is notconsidered as making a division of AB, this negative root is re-jected.
Prod. II. In a given Triangle ABC to inscribe a Square.
(Pi- 1. Fig. H.)
Suppose it to be done, and let it be EFIIG. From A letAD be perpendicular on the base BC, meeting EF in tv.
Let BC = a, and AD = p, both of which are given becausethe triangle is given. . Let AK bo assumed as the unknownquantity, because from it the square can easily he constructed;and let it be called x. Then (K 1) — FG —) l-T— p — x.
On account of the parallels EF, BC, AD : BC :: AK : F.1’;that is p : a : : x: p — x, and p* — px ~ ax, which equation, beingP‘‘
resolved, gives x=-.
p+a
Therefore x or AK is a third proportional to p~\-a and p, andthe point K being found, the construction of the square is sufli-ciently obvious.
Definition of Lines by Equations -
(38.) When curve lines are considered algebraically, they aresupposed to be produced hv the extremity of one straight line, asPwl moving in a given angle, along another straight line .115 in agiven position, which is called the base. (Plate 1. Eig. 19.)
This branch of the application of algebra to geometry, was in-troduced by Descartes , in his Geometry, winch is the new orhigher geometry, and respects the nature and properties of curvelines.
2. The straight line PM, moving along the other, is called anordinate, and is usually denoted by y.
3. The segment of the base AP"between a given point in it A,and an ordinate PM, is vailed an absciss, with respect to that or-dinate, and is usually denoted by x. The ordinate and abscisstogether are called co-ordinates.’
4. If the relation ot the variable absciss'and ordinate A P andPM be expressed by an equation, which besides ;r and y containsonly known quantities, the curve, described hv die 'motion ofthe point M in the ordinate, is called the locus of. that equation,and the equation is called the equation of the curve.
5. If the equation he finite, the curve is called algihrtiical.N. B. The terms geometrical • and algebraical, as applied to run elines, are used in different senses by different.writers; there areseveral other classes of curve-, besides what are here called alge-braical, which can be .treated of mathematically, tad r,m t>ymeans of algebra.
ti. 'l'he dimensions of such'equations are estimated from thehighest sum of the exponents of x and y in any term. Tim-, theterms a 4 , .r't/, x'bf 1 , xy*, if, arc all of the same dimensions.
7. Curve lines are divided into orders, from the dimensions oitheir equations, when freed front fractions and surds.
Ike