DEDICATION.
THAT You may Long enjoy Perfect Health and Feli-
city, and fee all Your Endeavours for our lnterefi andTranquility crown'd with Succefs, 1ha11 ever be the Sincere
\Vifhes of him who Humbly begs Leave to be, with theGreatefi Submiffion and Ref pee!,
SIR,
Your Moft Obedient,
Moft Dutiful, and
Moil: Humble Servant,
EDWARD OAKLEY.
•
a 2 PREFACE..
DEDICATION.
THAT You may Long enjoy Perfect Health and Feli-
city, and fee all Your Endeavours for our lnterefi andTranquility crown'd with Succefs, 1ha11 ever be the Sincere
\Vifhes of him who Humbly begs Leave to be, with theGreatefi Submiffion and Ref pee!,
SIR,
Your Moft Obedient,
Moft Dutiful, and
Moil: Humble Servant,
EDWARD OAKLEY.
•
a 2 PREFACE..
. _, p C E. ..
HE following Sheets are Collcll:ed, and Defign'd, for the
Affiftance and Inftrull:ion of fuch Perfons who delight in,
or are willing to proceed after a regular Manner in theScience of ARCHITECTURE.
There i no Occafion to make any Oration in Praife of this Noble ART;
the Eftirnation it bears, with the moft judicious Part of Mankind, beingfufE.ciently known; and that it has been, and is Encouraged, Studied,
and Pralhfed, by the moft Dignified and Renowned.
We ought in a particular Manner to Celebrate the Memory ofthat GreatRefrorer of Ancient ANCHITECTURE (in this our H1e) Inigo JfJnes,
and the moft worthy, valuable and indefatigable Genius of Sir ChrijlopherWren; thefe have embellifh'd the Kingdom, which with the continuedLabours and lndufl:ry of the Noble and truly Worthy Profeffors of this
D1v 1NE ScrEN cE, the Right Honourable the Earl of Burlington, theHonourable Lords Hcrhert, and Bingley, &c. will leave to Pofrerity moftGlorious Examples of the Beauty and Harmony of Proportion andDecoration.
But as the ingenious .Ard.ft and Pracl:itioner was oblig'd to haveRecourfe to many Volumes, to find out the different Parts ofthe fameScience; I have, for their Advantage, extracted the mofr Material Preceptsfrom our befi: Authors, and reduced them to the Eafieft Pratl:ice.
I hope the Acknowledgment I 1nake, by Naming the Authors, fromwhom I have Elell:ed, will fufficiently clear me of the Imputation of aPlagiary; feei11g efpecia!ly, that I return to the Publick what I borrowed
of them, viz.
For the four firft PART s I am beholden to 'Palladio, Scam1Jzz.i, Vig·nolaFreart, Perrault, Bo[fc, Le Clere, 'Pozzo, and Sir Henry Wotton; and for th;laft pART to /llberti, '])a Vinci, omatius.) Jincl Audran: 4nd as myCollelting from thefe Great Men, 1s no mbre than what themfelves havedone from each other, for the Benefit of the Publick; I wifh the Prefentand Future indu!l:rious Practitioners, and the Curious and ImpartialReaders, may receive a general Satisfaltion and Benefit from thefe myEndeavours for their Advantage.
Eftares furvcy'd, Defigns made, and Eftirnates calculated, for Building or Repairs•Articles and Contralls for Agreements witb Workmen fairly·drawn; Artificer;Works infpe8:ed, meafored, and Biils adjufted : And all Affairs relating to Buildingcarefully managed, By
EDWARD OAKLEY.
. _, p C E. ..
HE following Sheets are Collcll:ed, and Defign'd, for the
Affiftance and Inftrull:ion of fuch Perfons who delight in,
or are willing to proceed after a regular Manner in theScience of ARCHITECTURE.
There i no Occafion to make any Oration in Praife of this Noble ART;
the Eftirnation it bears, with the moft judicious Part of Mankind, beingfufE.ciently known; and that it has been, and is Encouraged, Studied,
and Pralhfed, by the moft Dignified and Renowned.
We ought in a particular Manner to Celebrate the Memory ofthat GreatRefrorer of Ancient ANCHITECTURE (in this our H1e) Inigo JfJnes,
and the moft worthy, valuable and indefatigable Genius of Sir ChrijlopherWren; thefe have embellifh'd the Kingdom, which with the continuedLabours and lndufl:ry of the Noble and truly Worthy Profeffors of this
D1v 1NE ScrEN cE, the Right Honourable the Earl of Burlington, theHonourable Lords Hcrhert, and Bingley, &c. will leave to Pofrerity moftGlorious Examples of the Beauty and Harmony of Proportion andDecoration.
But as the ingenious .Ard.ft and Pracl:itioner was oblig'd to haveRecourfe to many Volumes, to find out the different Parts ofthe fameScience; I have, for their Advantage, extracted the mofr Material Preceptsfrom our befi: Authors, and reduced them to the Eafieft Pratl:ice.
I hope the Acknowledgment I 1nake, by Naming the Authors, fromwhom I have Elell:ed, will fufficiently clear me of the Imputation of aPlagiary; feei11g efpecia!ly, that I return to the Publick what I borrowed
of them, viz.
For the four firft PART s I am beholden to 'Palladio, Scam1Jzz.i, Vig·nolaFreart, Perrault, Bo[fc, Le Clere, 'Pozzo, and Sir Henry Wotton; and for th;laft pART to /llberti, '])a Vinci, omatius.) Jincl Audran: 4nd as myCollelting from thefe Great Men, 1s no mbre than what themfelves havedone from each other, for the Benefit of the Publick; I wifh the Prefentand Future indu!l:rious Practitioners, and the Curious and ImpartialReaders, may receive a general Satisfaltion and Benefit from thefe myEndeavours for their Advantage.
Eftares furvcy'd, Defigns made, and Eftirnates calculated, for Building or Repairs•Articles and Contralls for Agreements witb Workmen fairly·drawn; Artificer;Works infpe8:ed, meafored, and Biils adjufted : And all Affairs relating to Buildingcarefully managed, By
EDWARD OAKLEY.
..
the
10,
the
eaties,
bcrredhisheoftnd
1vene
)tS
1m
ra'l1e1yvental1y
••••••••••••• • ························••+••···········
T H ECONTENTS
PART I.
Practical Geometry.
S E C T. i.
TrJ defcrihe Polygons, &c.
Pl te Fig, Page.
0 erecl: or let fall Perpendiculars I : 1>2,3,4: 1To draw a Line parallel to a given Line I : S : zTo 2,3,4: 1To draw a Line parallel to a given Line I : S : zTo TOc
TopreparetheStuffofwhichtheRailistobe
ade 5: 2, 4: 1sTo defcribethe Arch, or Mould, for a Hand-rad to an Oval Stair-Cafe 5 : 5, 6 : 16
To form the Arch-Mould to the Hand-Rail that fweeps two Steps, 5: 7to 14: 16,17
S E C T. 6.
To defcrihe Cavetto's, Cima, S,otia:, Eggs, .Anchors, &c.
T
T
o
defcribe Cavetto's
6: 1,2,3,4: 19To defcribe Cima's
6 : 5,6,7,: 19To deferibeScotia's
6:8910II : 19,20Todefcribe ap Ovolo in the Shape ofan Egg,and its fide Ornaments 6 12,13,14 zo
8 E C T. 7.
To
To
defcribe circular and oval Volutes, for Ionic Capit ls
7: l to 12: 21122
S E C T.
Two difft:rent Profiles of the T1,fca11 Order
Two difft:rent Profiles of the T1,fca11 Order
The C O N T E N T S.
S E C T. 8.
TO defcribe wreath'd ColumnsTo flute Columns and Pilafters
Sir Henry Wotton's Elements of ArcbiteB:ur
A Judgm
ent in General on all the Authors cited m the Parallel
PART II.
A
A
PraB:ical Treatife on the five Orders of ArchireB:ure
To find the Height reqt.iir'd of a Statue
Arcades of the five Orders or Figure elevated
10 to 15 : 0: 64
A Comparifon of the two Scales
made ufe of, viz. !v.fodules and Feet
o
The Proportions of the General Heights, and Pro1eB:ures of the Parts(_
o 64
belonging to the five Orders, the Entablatures of each being of the sr6, 17 : o: 65
height of the Column
A Profile of the Doric Pedeftal and Column
Two Profiles of the Doric Entablatures and Capital
A Profile of the Ionic Peddl:a! and Column
The Antique Tonic Capital
Modern Io11ic Capital
Ionic Entablature
Co,inthiao Pede/la!, Bafe and Sliaft of Column
Capi l
Cori11thia,, Entablature
Campofite Pedefral, Bafe and Shafe of Column
Compofite Capital
Compofite Entablature
lmpoCts and Arches to the Doric, Ionic, Corinthian, and Cornpofite
In the Ord,r; fallowing, the Entahlatures to each arc i •f
Column.
The T,Jca11 Order, with Impoft and Arch
Two diff rent Profiles of tht! Doric OrderIonic Order
Co,i11thi11n Order
Com
pofite Order
.Fr?ntif
iece of the
Tttfcan Order
Ditto ot the Do,ic Order
Ditto, Tonic Order
Ditto, Cori11thia11
7
!
ufri
a
tedFrontifpieces, and Columns
->JXdifferent Defigns of Windows ornamented
Two e11etia11
\an one Semi-circular headed) \Vindows
Mo ldmgs (ennch dand Plain) made ufe of to conftrucl: the Orders
Inte ·colu
Variety of Le ves, Rofes, &c. made ufe of to confrruB: Capitalsmnations, and of placeing Columns and Pi.l.ifrers
Vanety of Ballufters
0 l'o dimini{h Columns
l'o Pitch Pedements
En ichments for Freezes
Jome and C0Yinthia1t Pedefla!s
Pedefials for Statues, &c.
Ornarnen s of Fretts and Flowers
Compart1ments_ for Dom
es, Soffites of Arches, &c.
Of the P oportrnn and Cieling of RoomsCompart1ments of Pavemt:nts
' . Obelisks
b 2
Plate Fig. Page.
8 : 1, 2 : 21
I : r, 2, 1 : 24
0 0 25
0 0
57
Plate Fig. Page,
O o
59
60: o: 63
18, I 9 : 0 :
66
20 ° 67
21, 22: o : 67
23 o 67
24 ° 7
2 5 °
26 o 677
27
° 7
28 o 7
29 o 68
30 ° 68
3 r o 68
32 o 68
Orders o 68
33
the Hcit,ht ,f the
34 0 68
35, 36: 0: 68
37 o 68
38 0 68
39
0 68
40 0 68
41 0 68
42 0 68
43 0 68
44 0 69
45 ° 69
46 0 69
47 0 69
48 0 70
49
o
i
} 50 0 72
51 0 72
52 0 72
53 ° 72
54 ° 73
55 ° 71
56 0 73
PART
The C O N-T E N T S.
PART III.
Plate Fig. Page,
A
A
Treatife on Stair-Cafes, with the different FormsOf Irregularities, and the Method to reform them 5758
°
0
7475Three gra Stair-Cafes 59 0 76
PART IV"
Pra8ical Perfpellive.
Plate Fig. Page.
EX p LI<;: AT10N of the Lines of the Plan and orizon, €.:Jc. 6o
O 79
To delineate a_Squar , Oblong? or double Squa1:e, mperfpeltive 61 :1,2,1 : 8o' 81
s of Sq ares, with the1 -Elevauons, and of delrneating in Per•} .
fpelhve without Occult Lmes 62 • 4 5 : 8r, 8
Ditto, and t? defcribe Circles_in Perfpe8:_ive1 63 : 6 7 : 8z
The Projelbon of aPeddlal m PerfpeB.1ve
64 o
• 81
Attick Bafe, Ditto 6S o 8
Shaft of a Column 66 o 84
Doric C:apital . 67 o 84
Corinthuzn Capital 68 o 84
'J)oric E_ntablature . 69 o 85
Corinthian Entablature, Capital, and part of the Column 70 0 85
To defcribe the T1tfcan Order compleat 71 72 o : 86
'71
The Compofite wreath'd Column compleat
o
86
Tofind on Geometrical Bodies, the Geometrical Places of their}
7 4 0
Lights, Shades and Shadows
PART V.
On the Proportion if H1tman Body, &c.
Plate, Page.
L
L
EON, Baptift
aAlberti of Statues
75 88'Jo, Patil Lomatitts of Statues. On the External Parts of Man's Body oo 98The Proportion of a Body of Seven Heads in Height oo 100A Body of Eight Heads in Height oo 102A Body of Nine Heads oo 10JA Child of Six Heads oo 105A Child of Four Heads oo 106The Rule ofthe Defign of Natural Motion, E5c. by Leo D'vinci, &c. 76,77,78,79,80
: 107Girard ANdran, on the Proportion of Human Body 81 ro 93 : 108The Farnefian Hercules, is 7 Heads, 3Parts, and 7 Minutes in Height 81
Ditto, Side and Back . 82, 81
An E,gyptian Term, 7 Heads, 1 Part, and 7 Mtnutes 84
Ve111u Aphrotlite.r, 7Heads, 3Parts 85
Back, and other Side of D_itto 86
.
Four Views of Apollo Pythzus, 7 Heads, 3Patts, and 6 Mmutes
87, 88, 89, 90
A Boy of Five Heads · 91
The Parts of the Face, of an Antique f7emts, meafured in the fame Bignefs
} 92
as the Originals
lhePares of the Face of an Apollo, Ditto 93
PART
1ge.
8800ozOJo,060708
T
PART I.
A.Treatife of Practical Geometry.
S E C T. I.
To dejcrihe Polygons, &c.
PROBLEM I. PLATE I, FIGURE t
To erell a 'Perpendicular upon the middle of a given right Linc.7
DM I T C be the point propofed in the middle of the line AB.•Upon the given poinc C, defc1•ibe at pleafore the femicircle DE,upon the points D&E, make the fe8:ion I, from the point C, drawthe line demanded CO, thro' the Se8:ion I. this line CO will beperpendicular to the line given AB, and ere8:ed upon the pointpropofed C,
.l'Ro . -:i. FIG. '.l.
To erei1t a Perpendicular, upon the Extremity Of' (4 ,;.,,.-z_;,.,,,3ADM IT a b, the line given, and b the point or end on which the perpendicularis to be raifed.
From the point b, on the line ab, make five equal divifions towards a, upon thellOint b, with four of thofe divjfions as bd, defcl'ibe the arc f, upon the point c, withfive divifions as b e, defcribe the arc g, from the point b thro' the interfelbonfg, drawthe line b h, this line b h will be perpendicular to the line a b on the end b.
PROB. -FIG, 3·
Another Way to er ell a 'Perpcndicular upon the Ext emity ofa given Line;ADMIT abthe given Line, and a, the point propofed.
Upon the point a, defcribe the arc cf, with the radius ac, from the point c to.:Wardsfon the arc cf, make the points d&e, upon the points d&e, defcribe the arcsg_&b, from the point a, thro' the interfefrion g h, draw the line a i, which is rhe perpen-:dicular propofed.
pRO B. 4· F 1G, 4·
To let fall a Perpendiculm· upon a given Line, from a Point without the LineADMIT C be the P.oint from which a Line is to be let fall perpendicular to A B.. {!P0n the given point C, defcribe at pleafure the arch DB, cutting the line A B.1 th_e pomts &E, upon the points D & E, p1ake the Seltion F, draw the line CF, andt elme COw_1ll bethe line required.
1ge.
8800ozOJo,060708
T
PART I.
A.Treatife of Practical Geometry.
S E C T. I.
To dejcrihe Polygons, &c.
PROBLEM I. PLATE I, FIGURE t
To erell a 'Perpendicular upon the middle of a given right Linc.7
DM I T C be the point propofed in the middle of the line AB.•Upon the given poinc C, defc1•ibe at pleafore the femicircle DE,upon the points D&E, make the fe8:ion I, from the point C, drawthe line demanded CO, thro' the Se8:ion I. this line CO will beperpendicular to the line given AB, and ere8:ed upon the pointpropofed C,
.l'Ro . -:i. FIG. '.l.
To erei1t a Perpendicular, upon the Extremity Of' (4 ,;.,,.-z_;,.,,,3ADM IT a b, the line given, and b the point or end on which the perpendicularis to be raifed.
From the point b, on the line ab, make five equal divifions towards a, upon thellOint b, with four of thofe divjfions as bd, defcl'ibe the arc f, upon the point c, withfive divifions as b e, defcribe the arc g, from the point b thro' the interfelbonfg, drawthe line b h, this line b h will be perpendicular to the line a b on the end b.
PROB. -FIG, 3·
Another Way to er ell a 'Perpcndicular upon the Ext emity ofa given Line;ADMIT abthe given Line, and a, the point propofed.
Upon the point a, defcribe the arc cf, with the radius ac, from the point c to.:Wardsfon the arc cf, make the points d&e, upon the points d&e, defcribe the arcsg_&b, from the point a, thro' the interfefrion g h, draw the line a i, which is rhe perpen-:dicular propofed.
pRO B. 4· F 1G, 4·
To let fall a Perpendiculm· upon a given Line, from a Point without the LineADMIT C be the P.oint from which a Line is to be let fall perpendicular to A B.. {!P0n the given point C, defcribe at pleafure the arch DB, cutting the line A B.1 th_e pomts &E, upon the points D & E, p1ake the Seltion F, draw the line CF, andt elme COw_1ll bethe line required.
2 Praciical Geometry. Part I.
pRO B, 5· FI G, 5·
Through a given Point to draw a Line parallel to a given Line.LET A be the given point through which a line is to be drawn parallel to the
line BC,
Draw at pleafure the oblique line A D, upon the point A, defcribe the arc DE, upon
the point D, defcribe the arc A F, make the arc D G, equal to the arc A F, Draw the
line required MN, thro' the points AG, which is the line required.
pR O B. 6. FI G, 6.
To divide a given right Line into any Number of equal 'Parts.LE T AB be the line propofed to be divided into fix equal parts.
From the point A, draw at pleafore the line AC, thro, the extremity B, drtw the
line B D,parallel to the line AC, from the points A & B, and along the lines AC & B D,
Carry any fix equal Parts, viz. efghik, along the line AC, r q p o11 m along the line
B D, draw the lines e n, f o, g p, b q, ir, then the line AB will be divided into fix
equal parts at the Sections S, T, V, X, Y.
PROB. 7· FIG, 7·
To draw a fiiral Line ahout a given Line.LET I L be the line about which the fpiral line is to be defcribed.
Divide half the line IL, into as many equal parts as there are to be revolutions;
Example tr>make four Revolutions.
Divide the half B I, into four equal parts DC E G I, divide atfo BC into two equal
parts in A, uqon the point A. dPfc,·ibe the femicircles BC DE, FG, HI, upon the pointB, defcribe the femicirdes CD, E F, G H, I L, and you will have the fpiral required.
PROB, 8. FIG. 8.
To make an eq11,ilateral Triangle up(/n a given Line.
LET A B be the given line upon which the triangles is to be confrrull:ed.
Upon the extreme point A, with the radius A B, defcrib the arc B D, upon the
extremity B, with radius BA, defcribe the arc AE, from the interfell:ion G, draw the
lines CA, CB ; A BC wi11 be the triangle required.
pR O B. 9· FIG. 9·
To make aTriangle whofe Sides arc equal to three Lines given.
LET A, B, C, be the three lines given.
Draw rhe line DE, equal to the line AA, upon tbe point D, with the radius B B
defcribe the arc GF, upon the point E, with the radius CC, defcribe the arc HI, fro
the ioterfeltion O, draw the lines OE, 0 D, the triangle DE O, will be compofed of
three fides, equal to the three fides given AA, B B, C C.
PRoB. Io. FIG, 1o
To make a Square upon a given right Line.
LET A B be the given line.
Erea: tl1e perpendicular A C, upon the point A, defcribe the arc B C, upon the
points B & C, with the radius A B, make the ftll:ion D, from the point D, draw the linei
D C, DB; A BC D is the fquare which was to be conftrulted.
PRos.
2 Praciical Geometry. Part I.
pRO B, 5· FI G, 5·
Through a given Point to draw a Line parallel to a given Line.LET A be the given point through which a line is to be drawn parallel to the
line BC,
Draw at pleafure the oblique line A D, upon the point A, defcribe the arc DE, upon
the point D, defcribe the arc A F, make the arc D G, equal to the arc A F, Draw the
line required MN, thro' the points AG, which is the line required.
pR O B. 6. FI G, 6.
To divide a given right Line into any Number of equal 'Parts.LE T AB be the line propofed to be divided into fix equal parts.
From the point A, draw at pleafore the line AC, thro, the extremity B, drtw the
line B D,parallel to the line AC, from the points A & B, and along the lines AC & B D,
Carry any fix equal Parts, viz. efghik, along the line AC, r q p o11 m along the line
B D, draw the lines e n, f o, g p, b q, ir, then the line AB will be divided into fix
equal parts at the Sections S, T, V, X, Y.
PROB. 7· FIG, 7·
To draw a fiiral Line ahout a given Line.LET I L be the line about which the fpiral line is to be defcribed.
Divide half the line IL, into as many equal parts as there are to be revolutions;
Example tr>make four Revolutions.
Divide the half B I, into four equal parts DC E G I, divide atfo BC into two equal
parts in A, uqon the point A. dPfc,·ibe the femicircles BC DE, FG, HI, upon the pointB, defcribe the femicirdes CD, E F, G H, I L, and you will have the fpiral required.
PROB, 8. FIG. 8.
To make an eq11,ilateral Triangle up(/n a given Line.
LET A B be the given line upon which the triangles is to be confrrull:ed.
Upon the extreme point A, with the radius A B, defcrib the arc B D, upon the
extremity B, with radius BA, defcribe the arc AE, from the interfell:ion G, draw the
lines CA, CB ; A BC wi11 be the triangle required.
pR O B. 9· FIG. 9·
To make aTriangle whofe Sides arc equal to three Lines given.
LET A, B, C, be the three lines given.
Draw rhe line DE, equal to the line AA, upon tbe point D, with the radius B B
defcribe the arc GF, upon the point E, with the radius CC, defcribe the arc HI, fro
the ioterfeltion O, draw the lines OE, 0 D, the triangle DE O, will be compofed of
three fides, equal to the three fides given AA, B B, C C.
PRoB. Io. FIG, 1o
To make a Square upon a given right Line.
LET A B be the given line.
Erea: tl1e perpendicular A C, upon the point A, defcribe the arc B C, upon the
points B & C, with the radius A B, make the ftll:ion D, from the point D, draw the linei
D C, DB; A BC D is the fquare which was to be conftrulted.
PRos.
D,
ne
1at
int
he
he
be
IC>
Seel:. I.
PraElical Geometry.
PR OB-11• F I G. 11.
To make a regular 'Pentagon upon agiven right Line,
L
L
ET A B be the line given,
Upon the extremity A, and with the radius A B, Defcribe the arc B D F, Erell:
the perpendicular AC, Divide the arc, into five equal Parts ID L MB, Draw the
line AD, divide the bafe A B, into two equal parts in O, Erecl: the Perpendicular OE,
upon the Interfeaion E, with the radius E A, Defcribe the circle A B F G H, Carry round
five times, the line A B, in the circumference of the circle, and a regular equiangular
equilate1·al Pentagon, will be compleated,
PRoB. 11. F1a. 11.
To make aregular Hexagon upon agiven right Line.
LET AB be the line propofed.
Upon the extremities A& B, and with the radius AB, Defcribe the arcs AC, BC,
upon the Secl:ion C, Defcribe the circle ABE F G, Carry fix times the line given A B, in
the circumference, and you will have a regular Hexagon ABE GF D, upon the given
\ine AB.
PRoB, 13. F1G. 13.
Upon agiven right line to dcf,:ribc any 'Polygon from an Hexagon to a Vode,agon
L
L
ET AB be a line upon which an Hexagon, Heptagon, or oaagon, E!c. is to
be made.
Bife8: the line AB in the Point 0, erea the perpendicular O I, upon the Point B
defcribc the arc AC, oint O with the radius O B, defcribe the arc BP, Upon rbe
point F with the the radius FA, dercribe a Circumference, and it will contain the line
giveA A B, fifteen Times. A11dJo aifo fo;-any other Polygon.
PROB. 15. FIG-15.
To find tbe Center of a given Circle.
LET AB C be the Circle propo[ed, whofe Center is to be found.
Draw at Pleafure the right Line AB, terminating in the circumference A BC,
Bilea the right line A B, by the Line DC, Bife8: a\fo th:: Hne CD in the Point F, the
l'oint F will be the center of the Circle required ABC.
B PR.OB.
IS..
4 PraEtical Geometry. Pare I.
pROB, I6. FI G, I6.
To defcribe an Oval upDn agiven Length.
LET AB be thegiven length upon which the Oval is to bemade.
Divide the line A B, into three equal parts AC DB, upon the Points C &D,
with theradius CA, D:!fcribe the circles AEF, BE F, upon the interfe8:ions B &. F,
and with the diameter EI, asaradius, defcribe thearcs I I;I, ,OP; A IH BP O will be
the Oval requir'd.
PROB, 17. FIG, 17.
To find the Center and the t•wo 'JJiameters of an Oval.
L
L
ET AB CD be the Oval propofed whofe Cente1· and Diameters are to be
found.
Jn the Ovalpropofed ABCD, draw at Pleafure thetwo parallellines, AN, HI, Bi
feB: th lines AN, HI, in the points L&M, Draw the line PL 1v1O, BifeB:it in E,
and thePoint E will bethe center. Upon the point E, Defcribe at pleafure the circle
FGQ, cutting the Oval in F&G, tl1ro' tbe.interfeaions F&G, Draw the right line
FG, BifeB: it in R, Draw the greatefr diameter B D, tl1ro'the Points ER, Thro' the
center E, Draw the leaft diameter A EC, parallel to the line FG, and what was
propo[ed will beeffell:ed.
pROB, I8. FIG, l8.
'lo defcrihe an Ellipti,k Arch by the Tramel, the Length and Height being given.
L
L
ET ABC i reprefent the Tramel, theleg C i being at right angles with the
head AB, in each there is a groove (as reptefcnted in the midft of each by the
firong blacklines) for the pins e,&f, which are fafl:e_ned to the rule D M, ofalength
gl·eater than iK, the pins e&f, mull befuc_tat Cuch D1_ftance, that when a pencil, E1c.
is put duo' ahole at g, the length e_g 1s equ l to 1 K, t e half of the bafe line of
the arch, and tbelength f g equal to 1H the height the archlS torife.
Operation.
Fix the Head oftheTramel AB, ontheJength ofthe arch K L, and the pencil point
g, at the point K, and the pins f e in th grooves A & i_C, with.one hand 1nove
the pencil g, and with theothe gu1d the p ns f,& e, rntheir refpeB:,ve grooves, till
.
the pencil g comesto L, which w1ll efcribe the required arch KH L.
S E C T. 2. p L A T E J.
To defcriheArches, O·vals, &c. by the lnterfellionofright Line .
PRoB, 19. Fra: 19.
To defcrihc a Gothick Arch re'llerfe by Interfa8ion of right Lines:
L
L
ET a, h, bethe bafe ofthearch propoied, and e, d the height required.
Drawthe line e, c, perpendicularto the Line a b, from themidft e, double
tothe heightpropofed e d, from the extremities a &b, draw the lines a c &
b c, divide the lines a c & l, c each into an equal Number of equal Parts at
pleafure (the greater the number is, the exalter will the work be) admit 18, thenif
frreigbt
• Seer. 2 •
Prafiical Geometry.
[.
fheight lines are drawn f
rom the Points of divifion r, 2, 3, 4, E:fc. bfthe line a c to the
corl'cfpondent points of divifion 1, 2, 3, 4, E§c. of the line
c b, the points of inter"!
fi:cl:ion will be in the arch required.
PnoB: '.lo. Fie.
-10
),
To dcfcribe a Segment of a Circle hy Interfellion, &c.
PROCEED as in the Gothick arch reverted, and the fegment will be complefed.
Je
F, To find the different Compreffure or Tbruft of Arches according to their Height• whereby the thicknefs of walls or piers are found capable to fupport the fubteodiog arcl1oDivide tile Segment ad b, into three equal parts, as a f, f g, & g b, Continue the occultline g b, to h, fo that b h be equai to b g, upon the point b let fall the perpendicular bk,
•wh_ich is the infide of the wall required, thro' the point hdraw the line i l parallel to b "k,and bi is the thicknefs of tl1e wall or peer required. In the _fame manner proceed fot
be any other arch, as Fig. 21, 22, & 25.
or any other arch propofed.
i
PRon. 'lI. Fto. 'lt.
E, To dcfcribc an Elzipticle Arch to anj Widtb or Height propofed.
:le
ne
he LET a b be the width, uponthe points of extremi y and b, ralfe the perpe?cl_icula'.
11 c and b d equc1J ro the height propofed, draw die line c d parallel to a b, div1de die
as
line c d in half ate divide a c & /1 d c e & ed, each into the fame number of equal parts,
)
)
and draw the correfpondcnt incerft:cl:ing lines, according to the 19th Problem, and the arc
a eb will be defcribed.
PRoB, 22. Fie. 2-,.
he To defcribe the Goibick Arch by Intc,jeaion of right Lines;
he t
rth LE a b he the width, and f e the height prq,<>r,.fl_
Upon the extremities a & b, erefr the perpendiculars a c & b a, cad, c'iu..1 co
fc. lialf
the height propoftd, f e, draw the lines e c & e d, divide a c and b d, e c & e d; each into
of
the fame number of equal parts, and draw the correfpondent interfelting lines as before
dire8:ed, and the arch a e b will be defcribed.
N. B. If th Arch is requi
ed to be qui
tke, or _/latter on the Hanfe1 it is l11t leugtheJJinjt
or Jhortenzng the perpend1c1tlar Jines a c & bd.
'nt
ve PROB. 23.
FIG. -13.
'.ill
To dcfcrihe the Gothick Arch rnmpant.
DRAWth occult line a!., the horizontal width of the arch required, on the middfe
at I, ra1fe the perpendicular fe, upon the points a & g raife the perpendicularsa c
& g d, rn_ake g bequal to the height of the ramp, and draw the line a/;, make h e equal1? the height of the arch required, and a c & b d equal each to the half of he, draw thelines ce& ed, divide ac&ce, erl& d b, ea:ch into the fame number of equal parts dta:W
the corrcfpondent int<:rfcB:ing lines as before direfred, and the arch required will be defcri1
bed.
PR o B. '.l4. FI G. '.l4
To defcrihe the Elliptical Arch rampant.
DR !,'!
e Occul'. line,f. on th middle at g, , ire the perpendicular g J, upon tl,dP
,le & f raife the perpendiculars ac& fe, make f b equal to the height of the
tamp, and draw the r
& •me a b k
req ..uued d 1 h h .
raw t 1eline
, ma e t e eight a c & be, equal to the height of the arch
at be , ce, a·v·...1 1 1.
f Ipars
1 Ille t 1e mes a c, cd, & de, eb, each mto the fame num
df eq\ia ,
draw the Correfpondent interfeB:ing lines as before djreB:ed
arcrho.Icqu1redw1ll
be defcribed.
;ht , and che
f .RO BJ
6 Praflical Geometry. Part I.
l:'ROB, 25. FiG, 'l)•
To defarihe the Gothick Arch reverfe another Way. B
D
D
RA \V a b equal to the bafe intended, and c d parallel to a b, and of dillan_ce equaf
to the height of the arch required, and in length equal to the half of a, b, and proceed
as for Fig. 21, and the arch will be completed.
p ROB, 6. FIG, 16 & 30:
To defcrihc an Oval.
,,-, HE Tr.anfverfe and Conjugate Diameters being giveh, and bife8:ed in the midd!g\.I. a.c right angles, proceed as by Fig. 21. and the Ovals required will be defcribed.
pROB, 27. Fig. 'l7•
To gefcribe an Arch of equal Height to a Semi-circle, hut of a longc-1 '])ijlcnt;
A
A
D MIT c g d to be a Semi-circle, and a b the length required for an arch to rife, e;;;
qual to the femicircle, draw efparallel to a b; make e fequal to c d, and proceed
as for Fig. 25, and the arch required will be completed.
p R O 13. '.l 8. FI G. 28.
To deferibc an Oval Jmaller at one End than the other;
,
LE T the Tranfver!e and Conjugate Diameters be given, as a!J& h g, and bifeaing
each other in the middle, draw e c ;iad :fd parallel to bg, make f, d, equal to three
fourths of h, g, thro' the points .fh & dg draw the Jinesfe &d c, and proceed as in. Fig.
26 nd 30, and the Oval \,Vill be defcribed which was required.
PROB. '.l9. F10. '.l9.
To defari.he an Oblique Oval.
A
A
DMIT ab, a e, e f & f b to be the fides of a Rhomboid, withir. which is to be
infcribed an Oval.
Draw the Tranfverfe Diameter c d, parallel to a b & e f, and the Conjugate Diameter g b
parallel to ae& bf, and proceed as in Fig. 26, 28, or 30, and the Oval required will be
de!cribed.
•
j
J
1.-
' e-
eed
A
E
JOI
J'
C q. -•.•". •,
t,,... .... d. ..,.
B--"'
a::::---6:---I .III'··, ..
a,'---i.--1
mr
I[B ·E
a -t----1?.-f---+--1-_Jt•.
::r::·
DB]
c
I Xn
.... · ..····...... --... ·· .... ....
... ..· . ···-··· .....,. . ..
, .. ·. ../>.·>·::: ::: :.· .·.·:. ::.-:. ·-...·-.....··...\.. --······
.A. l3
E·.· ·••······D
.
.
B;
-';_:-..· 1••• ; • ./g;, ·)'"
·•.:.t;..:. ,tl•• C•
A 1 ll
··············1
·•. ··----···
1.-
' e-
eed
A
E
JOI
J'
C q. -•.•". •,
t,,... .... d. ..,.
B--"'
a::::---6:---I .III'··, ..
a,'---i.--1
mr
I[B ·E
a -t----1?.-f---+--1-_Jt•.
::r::·
DB]
c
I Xn
.... · ..····...... --... ·· .... ....
... ..· . ···-··· .....,. . ..
, .. ·. ../>.·>·::: ::: :.· .·.·:. ::.-:. ·-...·-.....··...\.. --······
.A. l3
E·.· ·••······D
.
.
B;
-';_:-..· 1••• ; • ./g;, ·)'"
·•.:.t;..:. ,tl•• C•
A 1 ll
··············1
·•. ··----···
XXII
be .
I
gb
be
.
J ........ ..
Sea:.)•
' PraElical Geometry. 7
S E T. 3•
p LA T E 2
To defcril,e Circles, Ovals, Rampant .Arches, &c: l,y the Inter:
feElion efparallel Lines, to defcrihe an Eltipjis: To defcril,eOvals, Rampant and Gothick .Arches, generated l,y Segment
_ of Circles
:eROB. I.
FI G. I,'
To defcrihe a Circle by paralld Lines.
D
D
E S C R I B E
the fquare be g n, equal to the Diameter of the Circle propofec1,
Draw the diagonals hog & eo11,
draw the diameters 1,
the interfofrion a, and at right angles with 11 b& ng, divide b 4 & 4 n, n 1&
o, 3 & 4, o, 2, thro'
r, 9, each into two equal parts, by the lines d.1, to, fr,
L7, at rjght angleswith each other, upon each angle of the fquare, on the fides fet 1-15 ofthe diameter, as
ha, bc, c. 'thro' which clraw the lines af,
gonal fet 1-7 oftheir length bog, ore o11, as
cm, v w, z t:f, upon each angle on each dia5,
6, 7, 8, thro' the points 1, L, 5, o, 2,I, 6, 'J, 3, i, 7, h, 4, t, 8, k, 1, Trace the Circle defired.
PROB. '2,
FIG. 2.
To defcribe an Oval,
DES CRIB E the Oblong v e, & eg, equal to tf1e Tranfverfe and Conjugate Diameters, and proceed as in the former, and the oval required may be rraced.
pRO B. 3; FIG-3·
To defcrihe a Rampant Arch.
THEBafe 4, o, 2, being given, raife the perpendiculars 4, b, & 2, e, equal to theheight of the intended arch, draw the line be parallel to the bafe, and proceed as
in the former, and the arch will be defcribed.
pRO B. 4· FI G. 4·
To defcribc anA8ual Ell.ipjiJ
·.
LE'r the Tran[verfc Diameter 4, o, 2, and the Conjugate Diameteq, o,
b1fecbng each other at right angles-. ,, be given,
With the Interval o, 4, upon the point 1, on the line 4, o, 2, make the points d, b, in the
Points d& bfix two pins or nails, &c. then with a /Iring encompafs db;, and by turning
this firing J db, of equal force about the points db, in foch manner that its fides remain
bent, will defcribe the Ellipfis 1, 2, 1, 4, J.
pRO B. 5· F I G. 5·
To defc,·ibe an Oval at openi
f
ngo
the Compafs.
ADM
! T 4, o, 2 to be the Tranfverfe,and 1, o,2 the Conjugate Diameters given;
.
With the Interval o, 3 or o, 1, on o, 4 and o, 2, make the points o d and o 1, draw
the Ime;, 2, and &om the point jraffe the Iine;, S, pe,·pendicular to l, 2, to inteofell ,, 4,
and the interval o, ; will bethe diameter 4 d and 2b, to defcdberhc rmall arches 6, 2, 7,
C 2
and
8 Pare I.frattical Geometry.--------------
artd 8, 4, 9, make 3, c, equal to d 4, and draw the line c d, divide c d in the midft bythe perpendicular ea, and where it interfe8:s 3, a, draw the line adg, and with the inter•
val a 3, defcribe the arch 7, 3, 9, and do the like for that below, and the oval will bed -
fcribed.
pRO B. 6. FIG. 6 ... To dcfcribe an Oval another Waj.
ON the line o , make the point cat pleaturc, wicb the interval J,c, fi·om 4too maicethe point d, and from z too make the point 'b, which will de1cribe the an,hes 6,2,7,
and 8, 4, 9, draw the line c d, which interfe8: at right angles bye, a, and from the inter-
fo8:ion 3, a, thro' the points d, b, draw the lines a, tl, q, and a, b, 7, and with the intervala, 3, defcribe' the arch 7, 3, 9, and do the like fo1• that below, and the oval will be genera..
ted.
PRoB. 7· F1G. 7.
To defcribe the Variations of Circles, Ovals and Rampants between the Jame.
'Parallels.
VI z. m, M, C, 'and H, h, F, are Parallels, r, 01 t 1 TOR; N,O, P, are the ira.nf..
verfe diameters of the Oval and diameter of the Circle; X,0, V the Coniugatediameter of the fmall oval, and 11, o; x, and 1, o, 2, the Coniugates oftbe two Ramps, H 2K R M OH, and m, u, K, P, h, o, m, are equal to one another on each fide K, k.
PRoB. 8. FrG. s·.
To defcribe a Rampant Arch between the Parallels H, F, and M, C, and ftorizThree given Points.
LE T the points given be H, K, M, Draw the line H, 0, M, on_the middle at O drawthe perpendicular OK, parallel to HF and MC, draw the line F KC, parallel toHo M, draw rhe line F O, make the point_ F x qua\ to_ z. 7 of F 0, draw a line from xto H, bife8: x 6:at right an les by 5, A ra1fe a perpcndicuhi.r from F H, on the point H;
and the interfea1on on the lrne 5, A, will be the center to the arch H x 2• From tlpoint A, thro' the point M, draw the line A, 2, on the line CM?n the oint M raife t :
perpendicular MB, m_ake M equal o HA: and frtn the po1_nt A draw a line to thepoint B which will give a cnJugate diameter t,r. Bife8: A, B, in the midft will giveTranfv:rfe diameter T, O, R. The Interfe8:ioo of T, R, and A, 2, will be rhe center 0the [mall circle to defcribe the arch 2 K RV, draw a line from B thro, the interfe8:ion Lwhich will determine the arch 2 K RV, and the cente1• B with the interval By or BMcomplete the ramp at M.
PLATE 3; PROB. 9· FIG, 9·
Another Way to defarihc a Rampant Arch between Parallels.
TH IS differs not from the former, except in finding the conjugate diameter t, 0 r•
which is found by bife8:ing at right angles HK, by the perpendicular SA, whichinterfefrs KA and H A, at the point A? A K is qual to. A H, :-Vith che interval .AK orA Itdefcribe the arch K t H, upon the rntcrfe8:1on L with the interval L K, de(cribe thearch K RV, and with the interval B V, defcribe the arch V M, which compleat-s the.ramp intended.
1V
8 Pare I.frattical Geometry.--------------
artd 8, 4, 9, make 3, c, equal to d 4, and draw the line c d, divide c d in the midft bythe perpendicular ea, and where it interfe8:s 3, a, draw the line adg, and with the inter•
val a 3, defcribe the arch 7, 3, 9, and do the like for that below, and the oval will bed -
fcribed.
pRO B. 6. FIG. 6 ... To dcfcribe an Oval another Waj.
ON the line o , make the point cat pleaturc, wicb the interval J,c, fi·om 4too maicethe point d, and from z too make the point 'b, which will de1cribe the an,hes 6,2,7,
and 8, 4, 9, draw the line c d, which interfe8: at right angles bye, a, and from the inter-
fo8:ion 3, a, thro' the points d, b, draw the lines a, tl, q, and a, b, 7, and with the intervala, 3, defcribe' the arch 7, 3, 9, and do the like fo1• that below, and the oval will be genera..
ted.
PRoB. 7· F1G. 7.
To defcribe the Variations of Circles, Ovals and Rampants between the Jame.
'Parallels.
VI z. m, M, C, 'and H, h, F, are Parallels, r, 01 t 1 TOR; N,O, P, are the ira.nf..
verfe diameters of the Oval and diameter of the Circle; X,0, V the Coniugatediameter of the fmall oval, and 11, o; x, and 1, o, 2, the Coniugates oftbe two Ramps, H 2K R M OH, and m, u, K, P, h, o, m, are equal to one another on each fide K, k.
PRoB. 8. FrG. s·.
To defcribe a Rampant Arch between the Parallels H, F, and M, C, and ftorizThree given Points.
LE T the points given be H, K, M, Draw the line H, 0, M, on_the middle at O drawthe perpendicular OK, parallel to HF and MC, draw the line F KC, parallel toHo M, draw rhe line F O, make the point_ F x qua\ to_ z. 7 of F 0, draw a line from xto H, bife8: x 6:at right an les by 5, A ra1fe a perpcndicuhi.r from F H, on the point H;
and the interfea1on on the lrne 5, A, will be the center to the arch H x 2• From tlpoint A, thro' the point M, draw the line A, 2, on the line CM?n the oint M raife t :
perpendicular MB, m_ake M equal o HA: and frtn the po1_nt A draw a line to thepoint B which will give a cnJugate diameter t,r. Bife8: A, B, in the midft will giveTranfv:rfe diameter T, O, R. The Interfe8:ioo of T, R, and A, 2, will be rhe center 0the [mall circle to defcribe the arch 2 K RV, draw a line from B thro, the interfe8:ion Lwhich will determine the arch 2 K RV, and the cente1• B with the interval By or BMcomplete the ramp at M.
PLATE 3; PROB. 9· FIG, 9·
Another Way to defarihc a Rampant Arch between Parallels.
TH IS differs not from the former, except in finding the conjugate diameter t, 0 r•
which is found by bife8:ing at right angles HK, by the perpendicular SA, whichinterfefrs KA and H A, at the point A? A K is qual to. A H, :-Vith che interval .AK orA Itdefcribe the arch K t H, upon the rntcrfe8:1on L with the interval L K, de(cribe thearch K RV, and with the interval B V, defcribe the arch V M, which compleat-s the.ramp intended.
1V
-···- ------··----------· .......;
·· ---,
t'·1·'·
.
.
-···- ------··----------· .......;
·· ---,
t'·1·'·
.
.
.
.'
\
..__\
·,. ....
\ "··
...
...
P'
\
ect. 4. Praflical ·Geometry. 9
pRO .B. Io. FIG, Io.
Another Way to defcrihc aRampant Arch.
THISis performed by the fame method as Prob. 8. the difference is in BifetHng x Kat right angles by SA, interfelting KAat A, the point 2 is on the contrary fide of:k to that of Fig. 8. fo that the fmall arch 2RM beginneth at the point z, and terminatesat M, (the line SA does not interft:Et the horizontal lines Hrand MF at the centers Aand B) H11 is made with the interval VM: ofthe arch V MR 2•. The Ttanfverfe diame•
ter T OR, bifeEts the horizontal lines H rand M F. The centers foi: fmall circles are L, l,and the intervals to defcribe them are lHor L 2. K A is perpendicular to the line F K C,K A is the interval to defcribe the arch H rt tK 2.
pR OB. l I. F to. I I, I 1, I 3& I 4
T'fJ defcrihc Rampant At·ches.
THES E Rampants are generated on the foregoing principles, and therefore needs
not a repetition tif defcription ; they are the more perceptible by being andefcri•
bed. by Letters, and with the fame Letters and Figures as the former; only obferve Fig. t1·
at pleafure make t x equal to H L, and bife8: L x by S A, and upon the Center A defcribethROB, 15. Ft G. 19.
To dcfcrihe the Gothick Arch another Way. <
ADMIT A B the line on which the arch is to be defcribed;
Divide AB into three equal parts at the points C and D, from the points A and
B let fall the perpendiculars A E and BF equal to AD and B C, Thro' the points F C
and E D draw lines of length at pleafure, on the points C and D with the Incerval AC
or D B defcribe the arcs AG and B H, Upoh the points E and F with tbe Interval EH
or F G defcribe the arcs HK and GL, and the InterfeB:ion will complete the arch re-
quired, A, G, I, H, B.
PRoB. 16. F1 G. 10.
Todefcrihc the Gothicle Arch another Way.
DIVIDE A B ico three equal parts at C and D, upon the points A CD B, witl1
the Interval A D, defcribe four arcs, and thro' the Interfeaion E and the point D
draw the line E D H, thro' the lnterreaion F and the point C draw the line F C G, upon
the points C and D with the Interval C A or D B defcribe the arcs A G and B A, and up-