Appendix. tiple of a fourth proportional to the multiple so taken of 
240 
—"Y~" the first, and the second, and third terms. 
Fig. 194 
Pig. 195. 
Prop. XIII. Pros. 
. To find a mean proportional between two given lines 
AB, CD. 
Place BE = CD in a line with AB (Prop. 8.) Bisect 
AE in F (Prop. 2.). Make BG = BF a 1.) On 
F and G as centres, with a radius equal to FB, describe 
ares intersecting in H ; and the distance from B to H 
will be the fourth proportional required. 
It is manifest from the construction, that FBH is a 
right angle, and that’ H is in the circumference of a 
cirele of which AE is the diameter ; therefore AB: BH 
::BH:BE or CD. (Prob. 3. Sect. 4. Part 1.) 
Prop. XIV. Tueor. 
Having given two points in each of two straight lines 
to find the intersection of the lines. } 
We shall give an analytical solution to this problem. 
Indeed the whole theory might, with great advantage, 
be given under the analytical form. 
Let A, B be given points in the line AB, and C, D 
given points in ‘the line CD. Suppose the intersec- 
tion of the lines to be found, and 'that it is the point V. 
Draw V a on the other side of VC, so that the pe 
CVa, CVA may be equal; take V a=VA, and V=VB, 
and draw lines from A to a, and from B tod. .Then 
every point in CV will be equally distant from A and 
a, also from B and 4 (12, and 17 of Sect 1. meet 
hence ‘the distance aC = distance AC; and the dis- 
tance a D = distance AD; but AC and AD are known, 
because the points A, C, D are given; therefore the 
distances aC, aD, are also known, and consequently 
the point a is known. In like mantier it appears that 
the distances b C, 6 D are'equal to the known distances 
. 
2 
~ BC, BD; thus the point 5 is known. 
Draw BG rape Va,to meet Adin G. ‘The 
figure BGad, is evidently a Helogram ; therefore 
BG =ab,andaG=Bé; ews, and B 6 are lines 
ofa igteen length, because the points a,b, Biare known; 
therefore BG, a G are given distances; and ¢onsequently 
the point G is known. 
The triangles AGB, Aa V are evidently similar; 
hence AG: AB:: Aa: AV; thus, AV = aV is a fourth 
porate to three given lines; therefore it may be 
ound v4 Prop. 12. cages 0 in V is at known dis. 
tances from given points A, a, the position of the point 
Wisndeterthigeds: sorta it 
Construction. On C and D as centres, with radii 
equal to CA and DA, describe arcs to meet on’ the 
other. side of the line ata; also on the same centres, 
with radii equal to CB, DB describe arcs to meet in 
6. On Basa centre, with a radius equal to. a 4, describe 
an are, and on a 4s a centre, with a radius equal to B& 
describe another arc, toveut the former in G. Lastly, 
on A and a@.as centres, with a radius equal to a fourth 
proportional to. the distances AG, AB, A a, (found 
Prop. 12.) ‘describe arcs’ to intersect in’ the ‘point V, 
which will be the intersection of the lines AB, CD, as 
is evident from the analysis of the problem. 
Prop. XV. Prop. 
To divide the circumference of a cirele into four, and 
also into eight equal parts. 
GEOMETRY. 
This problem might be resolved by the problem for 
the bisection of an arc, but more. el a 
struction suited to the particular case. 
may be as follows. . a 
et ADB be a’semicircle, AD one fourth, and AE 
one eighthof the circumference. Draw the radii CD,CE, 
and draw EF, a tangent to the circle, meeting CD in F, 
Because CE bisects the are AD, it is perpendicular to 
the chord AD ; riow CE is also perpendicular to EF ; 
therefore EF is parallel to AD ; hence Shrvangieeh Die 
EFC are equal ; now: the angles ACD, CEF are-also 
equal, therefore the triangles ACD, CEF are similar ; 
and since AC = CE, therefore AD = CF... Join AF; 
and.in the right angled ‘triangle ACF, we have 
AF? = AC? CF, but CF? = AD? = AC? + CD? = 
2 AC*; therefore AF? 3 AC?: Now AB? = 4 AC?, 
therefore AF* = AB?—AC*, Place in the circle a 
chord BG to the radius, and join AG; then, be- 
cause AG? = AB?— AC%, it follows, that AF? = AG 
and AF = AG, . Hence this construction.» 
Determine the semicircle AGB as ustal, and on A 
and B as centres, with a radius equal to AG, the chord 
of two thirds of the semicircumference, describe arcs to 
intersect each other in F. | Place in the circle a chord 
AD equal to the distance from C to F, and'D will be the 
middle, of the are ADB... yJ’ yenve Filey a EET nhs 
Again, on F asa centre, with a radius equal to AC, 
describe an arc to cut the circle in E; and E will be the 
middle of the quadrant AD. (&) 
‘GEOMETRY, Desériptive, the name )to a 
branch of geometry, which has of late years been much 
cultivated by the French mathematicians, and in parti« 
cular by Monge, who may be regarded as its inventor. 
Its object is to represetit on a plane, which has but two 
dimensions, any object which has three; and which ad. 
mits of a strict definition. Descriptive -admi 
of a twofold application. First, it is employed by 
artists, to communicate to each other a knoy of 
different objects. Thus it furnishes'the means of con- 
ripper geographical and ical ‘charts ; also 
plans of buildings and machines, architectural’ designs, 
sun-dials, theatrical decorations, &c. In this point of 
view, it is the best method that can be employed’ to de 
scribe the forms and the relative’ positions of objects. 
In the next place, it serves as an instrument of research, 
by which we may discover every thing relative to the 
form, and the position of ‘the et dot of objects 
which admit of a rigorous definition. It is by the pri 
ciples of descriptive geometry, that stone-cutters, car- 
penters, ship-builders, and other artists, find the dimen- 
sions of the different of the works which ~ 
execute, in as far as t! dimensions result from the 
complete definition of the object. © if 
Descriptive formed an essential branch ot 
the education of the French: in the school of pubs 
lic works established at the beginning of the revolution; 
and it appears from the journal of ‘the Polytechnic 
school, that the scholars were, during a certain period 
of the course, employed six hours every day in’ — 
the numerous objects which were the ‘subject of 
studies. ‘The lessons give ¢ 
a treatise on the subject by gr entitled Geometrie 
Descriptive, printed in 1799. There is also a treatise 
" by Lacroix, entitled Essais de geometrie sur les plans et 
les surfaces courbes (ou Elemens de geometrie 
We have already treated this subject tinder the 
— Carpentry. SeeCarrentry, Part II 
fot rtwogy 
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