To make a plumme line or any pricke that you will in any
right lyne appointed.
Open youre compas so that it be not wyder then from the pricke
appoynted in the line to the shortest ende of the line, but
rather shorter. Then sette the one foote of the compasse in the
first pricke appointed, and with the other fote marke ij. other
prickes, one of eche syde of that fyrste, afterwarde open your
compasse to the wydenes of those ij. new prickes, and draw from
them ij. arch lynes, as you did in the fyrst conclusion, for
making of a threlyke triangle. then if you do mark their
crossing, and from it drawe a line to your fyrste pricke, it
shall bee a iust plum lyne on that place.
[Illustration]
_Example._
The lyne is A.B. the prick on whiche I shoulde make the plumme
lyne, is C. then open I the compasse as wyde as A.C, and sette
one foot in C. and with the other doo I marke out C.A. and C.B,
then open I the compasse as wide as A.B, and make ij. arch lines
which do crosse in D, and so haue I doone.
Howe bee it, it happeneth so sommetymes, that the pricke on
whiche you would make the perpendicular or plum line, is so nere
the eand of your line, that you can not extende any notable
length from it to thone end of the line, and if so be it then
that you maie not drawe your line lenger from that end, then doth
this conclusion require a newe ayde, for the last deuise will
not serue. In suche case therfore shall you dooe thus: If your
line be of any notable length, deuide it into fiue partes. And
if it be not so long that it maie yelde fiue notable partes,
then make an other line at will, and parte it into fiue equall
portions: so that thre of those partes maie be found in your
line. Then open your compas as wide as thre of these fiue
measures be, and sette the one foote of the compas in the
pricke, where you would haue the plumme line to lighte (whiche I
call the first pricke,) and with the other foote drawe an arche
line righte ouer the pricke, as you can ayme it: then open youre
compas as wide as all fiue measures be, and set the one foote in
the fourth pricke, and with the other foote draw an other arch
line crosse the first, and where thei two do crosse, thense draw
a line to the poinct where you woulde haue the perpendicular
line to light, and you haue doone.
_Example._
[Illustration]
The line is A.B. and A. is the prick, on whiche the
perpendicular line must light. Therfore I deuide A.B. into fiue
partes equall, then do I open the compas to the widenesse of
three partes (that is A.D.) and let one foote staie in A. and
with the other I make an arche line in C. Afterwarde I open the
compas as wide as A.B. (that is as wide as all fiue partes) and
set one foote in the .iiij. pricke, which is E, drawyng an arch
line with the other foote in C. also. Then do I draw thence a
line vnto A, and so haue I doone. But and if the line be to
shorte to be parted into fiue partes, I shall deuide it into
iij. partes only, as you see the liue F.G, and then make D. an
other line (as is K.L.) whiche I deuide into .v. suche
diuisions, as F.G. containeth .iij, then open I the compass as
wide as .iiij. partes (whiche is K.M.) and so set I one foote of
the compas in F, and with the other I drawe an arch lyne toward
H, then open I the compas as wide as K.L. (that is all .v.
partes) and set one foote in G, (that is the iij. pricke) and
with the other I draw an arch line toward H. also: and where
those .ij. arch lines do crosse (whiche is by H.) thence draw I
a line vnto F, and that maketh a very plumbe line to F.G, as my
desire was. The maner of workyng of this conclusion, is like to
the second conlusion, but the reason of it doth depend of the
.xlvi. proposicion of y^e first boke of Euclide. An other waie
yet. set one foote of the compas in the prick, on whiche you
would haue the plumbe line to light, and stretche forth thother
foote toward the longest end of the line, as wide as you can for
the length of the line, and so draw a quarter of a compas or
more, then without stirryng of the compas, set one foote of it
in the same line, where as the circular line did begin, and
extend thother in the circular line, settyng a marke where it
doth light, then take half that quantitie more there vnto, and
by that prick that endeth the last part, draw a line to the
pricke assigned, and it shall be a perpendicular.
[Illustration]
_Example._
A.B. is the line appointed, to whiche I must make a
perpendicular line to light in the pricke assigned, which is A.
Therfore doo I set one foote of the compas in A, and extend the
other vnto D. makyng a part of a circle, more then a quarter,
that is D.E. Then do I set one foote of the compas vnaltered in
D, and stretch the other in the circular line, and it doth light
in F, this space betwene D. and F. I deuide into halfe in the
pricke G, whiche halfe I take with the compas, and set it beyond
F. vnto H, and thefore is H. the point, by whiche the
perpendicular line must be drawn, so say I that the line H.A, is
a plumbe line to A.B, as the conclusion would.