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is prac ra sto of prot ne ide llected ES:

Whence,

The root of which is y=?
25
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pa
=CB, as required.

?), the breadth required. 2s Prob. 2. It is required to deter Having thou found the breadth, mine the side of a square inscribed the length may be obtained by dis in a given triangle.

viding the area by the bread!h; or Let A B C (plate I. fig. 10,) re-otherwise we have, hy tinding the present the given triangle, and values of y instead of x, EFGH its inscribed square. Put

bp the base A B=b, the perpendicu

- pix

y
lar CD-a, and the side of the

6
square G F or GH=x; then will
CI=CD-DI=(-X,

And, because of the similar tri-
angles ABC, GFC, it will be, as
AB:CD=GF:Cl; or,

bp—

px

whence b : a = x :

b
Hence multiplying extremes and or .... bpr-px?
means: we have

ba
b a=b=; which gives x2 - 6x = -;
whencear+ 6x= ba,

P

b
ba
the side

whence
ato

2 of the square required.

Prob. 4. Iu a right-angled trianCor. Hence it is obvious, that in sle, having given the lengths of

two lines drawn from the acute all triangles whose bases and

perpendiculars are constant, the side angles to the middle of the oppoof the inscribed square will be con- site sides, to find the sides of the stant also.

triangle. Prob. 3. Giving the area or space

Let ABC (Fig. 12) represent the of a rectangle, inscribed in a given proposed triangle, of which the triangle, to determine the sides of lines AD, CE, are given. Make the rectangle.

AD=a, CE=b, AB=2x, CB=2y, Let ABC (fig. 11,) represent the AC=25. Then we have given triangle, and EFGH the inscribed rectangle,

the

area of
which is given, and which let be Now, .. 4x2 + 16y2=462

x2 + 4y2= 12)
represented by a; make CD-

again.
p,ab=b, EF=x, and ID=Ytherefore 15y2=462-a?

4.x? + y=al;
then will cl=p-y; and by simi-

41202
lar triangles we shall have
AB : CD = EF:CI, or

15
b : P = X : py;

CB. whence px=-by,

And in the same manner we find and ... dya, the urc a.

62 2r 2

- AB. br

15
The first gives I=--
P.

=AC.

15 the second., I=

Prob. 5. In a riglit-angled trian

gle, liaving given the hypotheninse bp -- by Whence again

and side of the inscribed square, lo .

find the base and perpendicular. P bpy-by2 = pa;

Let ABC (Fig. 9) be the proposed

triangle, AC the given hypothevo-pg-

nuse, and BD the given square. 45

6

4x2 + y2=a?) Euc, 47. 1.

, and se and pogled

erally repro

and consq. 2y = 2, (442)

cle, of , and y, the en we

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2

Make AC = h, FD=DE=S, AB Add and subtract double the last =x, BC-=y.

equation from the tirst, and we Then we have

have x2 + y2= 12 (Eucl. 47, 1.) x2 + 2xy + y2 = n2 + 2pn and (x - 5):s=s:(y- s) by

- 2xy + y2 = n2 - 2pn, sim. trians.

the roots of which are, Whence xy - 52 — sy +s?=s2;

2 +y= V(n2 + 2pn) or · • xy=s (x+y).

and 2 — y= Vin? - 2pn). By adding double this to the first equation, we obtain

Whence again, x=(ne + 2pn)+ x2 + 2xy + y2 = 12+2s (x+y),

(12— 2pn=AB,

and or (x + y)2 - 2s (x+y)=h2;

y=v(n2+2pn)whence x+y=si Vina+s2).

V (122 2pn) = BC. Now x +y being known, make it Wherefore the three sides AB, = n, then we have

BC, AC, are determined. x2 + y2 = h2

In the second branch of the Apx + y =n

plication of Algebra to Geometry, Square the second, and subtract or that which respects the higher it from double the first, and we geometry, or the pature and proobtain

perties of curve lines, the nature 22 - 2xy + y2 = 222 - n?. of the curve is expressed, or deBy extracting, 2-y= V(212_n2). noted by an algebraical equation, Again • x+y=n;

which is formed as follows: a line therefore,

is conceived to be drawn to repren+1 (292_n)

sent the diameter, or some other

= AB; principal line of the curve; and n-v(2212)

upon this line, at any indefinite and

: BC, points, are erected perpendiculars,

which are called ordinates; and as required.

the parts of the first line cut off Prob. 6.-Having given the sum by ihem are terned abscisses. of the three sides of a right-angled Calling the abscis x, and its cortriangle, and the perpendicular, responding ordinate y, the known let fall from the right-angle upon nature of the curve, or the mutual the hypothenuse, to find the three relation of the other lines in it, sides of the triangle.

will furnish an equation, involvLet ABC (fig. 12,) be the propos: ing x and y, with some other let. ed triangle, of which the sum of

ter or letters which are known. the sides and perpendicular BD are And as x and y are common to given. Make the sum of the sides every point in the primary line, s, the perpendicular=P, AB=X, the equation, derived in this manBC=Y, AC=%. Then we have +y +% =S,

ner, will belong to every position

or value of the absciss and ordi. by the question 22 + y2=%?

nate; and may be properly con(Euc). 47, 1.) x :x=p:y, by sim. sidered as expressing the nature trians.; or my=p.

of the curve in all points of it, Now, add double this last equation and is usually called the equation to the second, then

of the curve. Hence every partix2 + 2xy + y'= x2 + 2p% 22 + 2xy + y2

cular curve will appear to have

3? - 2- +2, by transposing in the first equa from that of every other; either

an appropriate equation, differing tion and squaring.

as to the number of the terms, the Whence, 2 + 2p%= 52--23% + m2, powers of the unknown quantities or,

= AC. x and y, or the signs of the co2p + 2s

efficients of the terms of the equaNow & being known, make it tion. =n; then the second and third APPLICATION of Geometry to equations become

Algebra, is the converse of the 22 + y2 = n2

tirst of the two preceding cases. XY = pn.

It relates principally to the find.

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ing the roots of an equation by as its point of regression, or vertex, geometrical construction.

is uppermost, and the descending APPLICATION of Algebra and body must commence its motion Geometry to Mechanics. This is in it with a certain determinate found on the same principles as velocity. Varignon rendered the the Application of Algebra to Geo- question more general by investi. metry; and consists principally in gating the curve which' a body representing, by equations, the might describe in vacuo, so as to curves described by bodies in mo- approach through a given point tion; as in the theory of Projec- through equal spaces in equal tiles, &c.

times, according to any law of gra. APPLICATION of Mechanics to Maupertuis also resolved the Geometry, consists chiefly in the same problem, in the case of a use that is sometimes made of the body descending in a medium, the centre of gravity of figures, for de- resistance of which is proportiontermining the contents of solids ate to the square of the velocity. described by those figures.

Method of APPROACHES, a term APPLICATION of Geometry and used by Dr. Wallis, in his Álgebra, Astronomy to Geography, princi- to denote a method of resolving pally consists in the three followcertain problems, relating to square ing articles; viz. in determining numbers, &c.; which is done by by geometrical and astronomical tirst assigning certain limits to the operations the figure of the terres quantities required, and then aptrial globe; in finding the positions proaching nearer and nearer till a of places by their observed latio coincidence is obtained. This metude and longitude; and in deter-thod was invented by Dr. Pell, for mining, by geometrical operations, the solution of equations of the the positions of places that are not form x? — ay2 = 1; which pro. very remote from one another. blem was proposed by Fermat, as Astronomy and geography are a challenge to all the English maagain applicable to the theory of thematicians of his time; viz. to navigation.

find rational and integral values APPLICATION of Geometry and of u and y, in the above equation, Algebra to Physics or Natural Phi- for every value of a, except when losophy. For this application we are it is a complete square.. indebted to Sir Isaac Newton, APPROXIMATION, in Algebra whose philosophy may, therefore, and Arithmetic, is the method of be called the geometrical or mathe approaching nearer and nearer to matical philosophy; and upon this the quantity sought, when there application are founded all the is no method of obtaining the ex. phyico-mathematical sciences.- act value. Hence a single observation or expe APPROXIMATION to the Roots riment will often produce a whole of Equations. As there is no di. science. Having ascertained, by rect method of determining the experience, that the rays of lighi, rools of equations beyond those of by reflection, make the angle of the fourth degree, and even in incidence equal to that of reflec- those of the third and fourth del tion, we hence deduce the whole gree. being very laborious by the science of Captoptics. The case direct rules, mathematicians have is also the same in many other endeavoured to find methods of sciences.

approximating the roots : of these, APPROACH. The Curve of equa- Newlon's rule is the most popuble Approach, is of such a nature, lar, and is founded on the followthat a body descending by the sole ing principles : power of its own gravity approach It any two numbers, being subes the horizon equally in equal stituted for the unknown quantity times. This curve has been found, in an equation, give results with by Bernoulli, Varignon, Mauper- opposite signs, an odd number of tuis, and otbers, to be the second roots must be between these numcubical parabola, so placed that) bers.

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This appears from the property .248 =8+12F+612+13 of the absolute term, and from this

-2x=-4-2f
obvious maxim, that if a number

-5--5
of quantities be multiplied toge 23.2–5 =-1+10+6f2+13=0;
ther, and the sign of an odd num- therefore, tu tiod f we have the
ber of them be changed, the sign equation
of the product is changed. For f3+92 +10f-1=0,
when a positive number is substi- which is still a cubic; but since,
tuted for 2', the result is the ab-

from the nature of our substitution,
solute term of an equation, whose
roots are less than the roots of the must be less than 1, the cube of

it may be omitted as inconsider.
given equation by that quantity.
If the result has the same sign as able, and this will give
the given absolute term, then from f2 +3f-1=0, or
the property of this term, either

25
none, or an even number only,
have had their signs changed by

36
the transformation; but if the re.

-5+ V31
sult has an opposite sign to that of

6
the given absolute term, the sign
of an odd number of the positive which value of f, being added to
rools must have been changed. 2, will give an approximate value
In the first case, then, the quan-

of x.
tity substituted must have been But generally, when f is less
either greater than each of an than unity, all the higher powers
even number of the positive roots of f may be omitted, without any
of the given equation, or less than yery sensible error; which being
any of them; in the second case, done in the present instance, will
it must have been greater than give 10f-1=1, orf=0:1, and
each of an odd number of the po-

therefore r= 2:1 nearly.
sitive roots. An odd number of Now as f=0:1 nearly,
the positive roots must, therefore,

let

f=:1+g,
lie between them, when they give and substituting this value for fin
results with opposite signs. The the preceding equation, we have
same observation is to be extend.

fs

= 0.001 + 0.03g + 0.3g? + g ed to the substitution of negative

6/?=0.06 + 1:24 + 6g? quantities and the negative roots.

105 =1. + 10g
Let it, for example, be proposed

-1=1
to find an approximate value of x 73 +682 + 10f-1=
in the equation

0.061 + 11.23g + 6 3g? + g3=0.
23.2.X-5=0.

And neglecting gs and gʻ, for the Here we easily see, that one root reasons above stated, this equation is between 2 and 3; for these num.

becomes bers being substituted for x, will

0.061 + 11.23g = 0,

0.061 give, the one a positive, and the

org

= .0054.
other a negative result. The ruot,

11.23
therefore, is greater than 2 and Hence, then,
less than 3, and we may, there =f=l+g='1- .0054 = .0946,
fore, write a=2+f, or a=3-finearly; and, consequently,
and' by finding f, or an approxi x=2+f=2+.0946=2.0946.
mate value of it, in either case, This operation may be extended
will obviously also give an ap- to any length at pleasure, by sup:
proximate value of x, by adding posing again g=-.0054 +h, and
ihe value of f to 2, in the first inence finding a value of n, and
case, and subtracting it from 3 in so on: this last substitution, if
the second case.

worked out, will give
Suppose x=2+f, and subtitut.

X = 2:09455147. ing this value for x in the propos

Another method of approximated equation, we have,

ing towards the roots of equations

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3.8'2 =

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in general, is by the following for-froot, we are rather under than mula, first published by Barlow: over the real state of convergency;

Let 2" tax-17.xn- +241-3+ for generally, if the first supposi. &c.=w, by any equation ; tion be a simple integer, the first and zin+ax'n-+bxin–2+czn3+ approximation may be obtained lo &c. =v, at being an approximate three places, which, though it be value of 2; then

not always correct in the last (10-0)2x1 ==x+

place, yet by employing it as a (1 - 1)w + (n + 1) xnt new approxiniation, the next value (1-1)ux-+(1-3).xn-+ &c.;

of x will, in most cases, be found (W-V)22!

true to six or seven fignres, which or, x=x+

is a degree of convergency that (1-1) v+(n+1) xin + cannot be obtained by any other (n-1)axın-I+(1-3)bxın-2+ &c. rule that we are acquainted with,

The first formula being applica- at the same time that the operation ble to the cases in which w is is much more simple than by any greater than unity, and the second other rule whatever. when a' is less than unity.

An example will sufficiently ilThese formnlæ are general for lustrate the preceding remarks. equations of all dimensions, but Example. Find the value of x when reduced to particular cases, in the cubic equation they become much more simple : x3 + 3x2 + 3x = 130; thus, reducing them both into one, Assume a = 4, then by formula (1)

X'3 we have for

64
2x'3

128
48

48 (1.) Equations of the third Degree, 3x 12

130 (w--V)21 x=x+ w or v+2013+ axl2

= 124

306 deno. = 130

minator (2.) Equations of the fourth Degree,

(w/v)2X r='+

6
3w or 3v + 5x^4 + 3ax's +
b.x'2 - CI'

306) 24 (4:08 nearly.
(3.) Equations of the fifth Degree, Therefore now X=4.08.

(W-Vr' I=x' +

Hence, adopting this new value, 2w or 20 + 3x6 + 2ax'+ + we have

X'3 67.9173

135.83416 b.xs_dar

49.9392 And in the same way we might 3x =

ar'2 = 49.9392

19.24 find the particular formula answer.

E 130 ing to an equation of any other

= 130.0965

315.7738 degree.

= 130 Application of the preceding For-w-v= 0965 mula.-In order to this it may be

4:08 observed, that æe is an approximate value of x, found first by 315-77) •39372 (-001247. trial, as near the true root as pos- Whence, 4:08 —:001247 = 4:078753 sible, which in all cases ought to Answer. be true to the nearest integer. This root is true in the seventh Then substituting this value of a', figure ; and, in the same manner, a nearer approximate value of x the root of any other equation may will be obtained, which will in all be determined. cases double the number of figures. APSES, in Astronomy, are the Then, considering this as a new two points in the orbits of the plavalue of x', another still nearer nets where they are at their greatvalue of x will be determined; and est and less distance from the sun so on.

or earth; the former being called In stating the degree of approx. the higher apsis, and the latter the imation to be that of doubling the lower apsis : but the higher apsis number of figures in the assumed is more commonly called

the

4

2.x'3

3x2 =

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