YALE UNIVERSITY
LIBRARY

A
TREATISE ON THE

Construction of 0l&$$:
IN WHICH
THE PRINCIPLES OF THE PROJECTIONS OF THE SPHERE
ARE DEMONSTRATED,

THEIR VARIOUS PRACTICAL RELATIONS
TO
MATHEMATICAL GEOGRAPHY,
DEDUCED AND EXPLAINED:
SYSTEMATICALLY ARRANGED, AND SCIENTIFICALLY ILLUSTRATED
FROM TWENTY PLATES OF DIAGRAMS.
WITH
AN APPENDIX AND
COPIOUS NOTES.

BY ALEXANDER JAMlESON.

" Nothing will contribute more to the advancement of geographical studies,
than the construction of maps."
Encyclopedia Brita.vnica, p. 548, Fifth Edition.

JLcmDoiu
Printed by Darton, Harvey, and Co.
For C. L a v," Ave-maria Lane; Black, Parry, and Co. Leadenhall Street;
J. M. Richardson, Cornhilli Darton, Harvey, and Darton, Gface-
church Street; J. Mawman, Ludgate Hill; G. Cowie and Co. Poultry;
J. Booth, Duke Street, Portland Place; and W. Kent, High Holborn.

1814.

PREFACE.

X HE universality of geographical studies, ber
speaks their importance ; and the motto which the
author has chosen, contains his apology for offering
this volume to the public.
In a scienee that has outlived the Vicissitudes of
twa thousand years, and become splendid amidst
even the riot of barbarism, originality is hardly to
be expected. If, however, the compiler has successfully arc-
ranged and condensed the scattered researches of
various authors, and united the theory of ancient,
with- the practice of modern, discoveries, he has,
perhaps, done all that could reasonably be ex
pected. How far he has succeeded1 in- the attempt,
must be referred to the decision of the judicious
and candid reader.
As a work of this kind must depend upon- a
great many relations, geographical and mathema
tical, the first two sections very naturally introduce
the student to the third1. In this section, the* prin
ciples of the orthographic, the sterebgraphic, and
A 2

IV PREFACE.
the globular projections cf the sphere, are fully
demonstrated; and the last of these is investigated
in a manner entirely new, to prove its superiority
and admirable fitness in the construction of maps.
In the fourth section, theory descends to prac
tice; and as certain combinations are proposed to
be effected, the projections are handled in the form
of problems: by this means they are reduced to
much greater simplicity than the prolixity of per
suing the subject in numerous subdivisions would
have allowed. , ,
Mercatqr's Projection might have been blended
•with the former; but it was more analogous to the
plan of the work, to assign a separate section to
the principles and practical methods of so ingenious
an invention.
The origin and properties of the rhumb line,
with its usefulness in navigation, occupying the
sixth section, are treated as concisely as the nature
of the subject would admit. ;
The meridional, equatorial, and horizontal con-
structions of Maps, in the seventh; section, are siu-
gularly beautiful, and highly interesting, the sub
ordinate parts of the problems having been enriched
•with valuable elucidations.
The principles of developing a spheric surface
on a plane, are investigated in the eighth section ;
and the application of the developement of the
conic surface, in the construction of maps, pos-
sesses the rare qualities of simplicity and elegance,
with a nice approximation to truth.

PREFACE: V
The ninth section is of a miscellaneous nature;
unfolding numerous projections of particular maps.
These : constructions are presented in a popular
form, and include whatever appeared of essential
consequence in modern practice.
Having treated so fully of the orthographic pro
jection of the igphere, it seemed necessary to shew
its extensive application in the construction and
use of the Analemma — an instrument that will
solve many of the common astronomical problems— ¦
and: the tenth section has been allotted to these
subjects. '¦
Without pretending to contribute to the ad
vancement of geographical 'studies, the Appendix
will probably be deemed not the least interesting
portion of this volume, since the ingenious resources
which it discloses, are calculated to afford very
pleasing and instructive exercises.
The Notes and Illustrations with which the work
concludes, will be found peculiarly useful, as the
various historical and critical remarks which are
blended with them, have resulted from numerous
sources of information* some of which are difficult
of access. When the student has attentively studied the
first three sections, he should exercise himself in
drawing small planispheres, agreeably to the dif
ferent methods laid down in the fourth section.
Mercator's Projection may then be performed, or
the constructions in the seventh section, may fol
low, on a small scale, those in the fourth; and the

1fl ' PREFACE.
eighth and ninth sections, executed on the same
plan, will form an agreeable variety, and make
mathematical geography exceedingly interesting,
as well as intelligible and instructive.
But to facilitate the design of this volume, th<?
student should provide himself with the necessary
instruments mentioned in the work, and a book of
imperial drawing paper, about fifteen inches by ten,
in which the projections may at first be made from
scales, nearly the size of those on the plates ; and
the successive novelty disclosed in the methods he
shall have to employ, cannot fail to produce that
assiduity and solicitude for excellepce, which will
crown his performance with merited success.

2l, Wdlt Street, Oxford Street, London,
Noii. Ut, 1814.

TABLE OJF CONTENTS.

SECTION I.— Preliminary remarks on the nature" 'or1 the Terres-
'trial Globe, its circles, and their uses  .*.... .'. 1
Divisiofiof tlrese into great and small  .. 2
Of the axis arid poles  ....'.  '.'  3
Ofthe Equator or Equinoctial Circle .....'.'.-. .............. ib.
Of the Ecliptic 'Circle '  '..:  ..... -... 4
Of'th'e'meridians or circle's of longitude  5
Of the'&raieh meridian  .....'.;... .':  ib.
Of'the parallels of latitude  V..'.  .'. 6
Of latftude and Ibhgitu'de ......»........:.... . . .'.... ... '. . . ib.
Illustration of latitude  ib.
— —  '¦ — oflongitucle  ;....  ."  ft
Demonstration of particular longitude   ib.
Foundation of fables of longitude  . . 9
'Fable 'i. shewing the length of a degree o'f longitude for every
"degree of 1atrti*d«,m geographical miles  , 1Q
Table It. shewing the same for English statute miles ...... 11
Method of reducing degrees to miles, and •vice versa ...... 12
Cbmputation'o'f longitude 'in time  ib.
Of the Tropics ............'.!......'..  '.'.'., .'.-.'... 13
Of the polar circles   ib.
Of the colures .  14
O'f tlie hemispheres  ib.
Of the wooden horizon ...  ib.
Of the positions of the sphere   IS
1st. The parallel sphere  .../  ib.
Sdly.'The' right sphere  16
3d1y. The oblique sphere  ;  j"4,
SECT? ION ' tl. — Preliminary observations an maps and charts .. 18
O'f tlie distinction between a map and a chart  ib.
The cardinal points of the compass  1:9
Of riumb'enng'the degrees o'f latitude and longitude on maps 20
latitude ami longitude by inspection  ....,'. ....... 21

Viii TABLE OF CONTENTS. % Page.
Method of finding the latitude and longitude on maps  21
Division of the earth  23
- Plains  ih-
Mountains  '*•
Oceans 
Seas  •  "¦'
Baysand Gulfs   *'*•
Straits  •  *'*•
Lakes  • 2«
Kivers  •  •*• '"•
Continents  ,  -  l"-
Peninsulas  — ¦• 27
Isthmuses  Es
capes or Promontories  &>•
Islands    ¦••. 28
Of the objects that diversify the face of a country  ib.
SECTION III.— On the principles of the orthographic, the ste-
reographic, and the globular projections of the sphere . . 3t
Principles of the orthographic projection  •  33
Proposition I. The figure of a straight line, AB, is a straight
line in the projection  34
Prop. II. The figure of the projection of a circle is an ellipse,
when the plane of the original circle is inclined to the
plane of projection  ib.
Prop. III. Equal parts upon the surface of the sphere, are
not projected into parts either equal or similar, except, as
above, in the case of circles that are parallel to the plane
of projection  35
Prop. If. The diameter of a circle is the projection of a serni-
-circumference of that -circle, when the plane of that cir
cle is perpendicular to the plane of projection ,,  36
On the stereographic projection of the sphere  37
Proposition I. The projection of an arc, measured from the
pole, is equal to the tangent of half that arc ......... 3%
Prop, II. Every oblique circle not passing through jt he pro
jecting point, is projected into a circle ,«••• ........ ib.
Prop. Ill' The projection of all circles, parallel to the plane
of projection, will be concentric circles  Z%
Prop. IF. Every circle of the sphere, whether great or small.

TABLE OF CONTENTS. IX
¦* : flip.
is projected into a straight line, when the plane of that
circle, or a portion of the.saine, passes through the eye or
projecting point  39
Prop.F. The radius of the projection of any great circle, is
the secant of the angle,; between the plane of the circle
and the plane of projection  ib.
Scholia  40
Prop. VI.. To find the projection of the parallels of latitude . 41
Of the globular projection of the sphere ,.  ib.
1st. Assuming the point of view at a distance from the surface
of the sphere, equal lo the sine of forty-five degrees . .. ib.
Sc%, Assuming this point of view, equal to threerfourths.of
- the. radius of the primitive circle  43
3dly. Assuming, this point of view on the vertex of ap equila-
„.,.> teral- triangle, having for its base the diameter of the
primitive cirple   45
On the disproportion of the distances, between the meridians
and parallels in the stereograpbic projection .>.  , 43
On the equality of the distances between the meridians and
parallels in the globular projection  49
Illustration of a planisphere .,..  ; 51
SECTION IV. — Containing geometrical and trigonometrical pro-
-. jections of the sphere, on the plane pf a meridian and
the equator  ..... 52
PrPB.lem I.(.To draw the>circles, of longitude geometrically,-
r ,, _for a _map of the world, according to the globular. nrojec-
,.,-. tion of the spiere, on the,.p,lane of a,merjdjan ........ j ib.
Prob. A. Totlecribe a. 'circle through any point D, .in the
semi-circlp.AC B  ......... k 55
Tr-ob. II. Ta draw the circles pf latitude geometrically, /or a
s, map of the world, according., fo the globular projection
of the sphere upon the plane of the meridian  , ib.
Prob. HI. ¦ contains, another method of Problem I. . ,-.-. . . 57
Prob. IV. contains anqthep( niethqd of Problem II. ..,.,. 58
Prob. V. To project the circles of longitude for a map of the
world, on the plane of the ^meridian, according to the
globular projection of the sph.ere, from a trigonometrical
table of radii  /  ¦  60
Trigonometrical table of radii for the meridians  „ ib.

• it t ABLE OF C&Nf ENfS* Tap!.
Prob. B. illustrates the canon of this trigonometrical table of
radii  ..................... 61
Prob. VI. To draw the parallels T>f latitude from a trigo-
'mettical table of radii,: for the globular projection of the
: ''Sphere on the- planfe of a 'meridian' -...'  62
Trigonometrical taMeof radii, for the parallels of latitude . . ib.
Pkob. C 'Demonstration of tire canon on which this fable is
• : fourrded  -.  *  -.  63
Pkob: VII. To project a map of the world geometrically oft
the plane df the equator, according to the globular projec
tion of the sphere ''.V;.  ib.
Prob; VIII. On the plane of the meridian, aird by the stereo*
" graphic projection of the sphere, to draw the -circles of lon
gitude geometrically for a map of the world ......... 64
Prob. IX. To draw the tircles1 of latitude geometrically, for
•a map of the world; •on thephire of the meridian, and by
the-siereographic projection of the'sphere  6S"
Peotb. X; To draw the circles Of longitude according to the
stereograpbic projection Of the sphere, oh the plane of the
nreridratr, 'from a trigonometrical table of radii  68
Trigonometrical table 'of ra'dii for the meridians, constructed
from a canon of radii, as exhibited in Problem B  69
ProbvXI. To dra1* thfe circles of Jatitu'd'e stereographically
on the plane of'th1* meridian, from a trigonometrical
table -of radii .....•.•.-.•;..-.,.....;  $.
This tabic1 of latitudes Constructed from a fcahOn of radii, ai
exemplified in Problem C. ...........  .„., 70
1 Prob: XII.! To make a stereographic1 projection 'cf the
sph'eYe, Off the plane of the equator  ib.
A rulWor determining the size Of a globe, Upon which the
countries shall be- of the saihfe dimension^' as you' see thein
Jn any mapof the wo'rld"  71
SECTION V.— The principles of Mercator's Projection, with a
' - • ¦ copious illustration Of this projection, anil Problems ex
hibiting this map constructed— first, from a table of me
ridional parts ;— and, secondly, geometrically;— and a
method of enlarging or reducing maps  72
Proposition I. Shewing that a degree of latitude is to a degree
of longitude, as radius is to the cosin-e Of latitude ..... ft

TABLE OF COiNTENTS. xl
Pajs-
Prop. II. Determines the length of, an arc of the meridian,
corresponding to any gi vera latitude  73
Table of meridional parts for every degree from Wo 84%
north or south latitude  ..;  75
Hemarks on Mercator's Projection  7 s
Prob. I. To draw tlie meridians and parallels for Mercator's
. Chart of .the World ..  77
Of the construction of a diagonal scale, by which to draw the
parallels of latitude for this map  78
Of the meridian, line on Gonter's Scale  73
Of drawing the parallels of latitude in this map  .. SO
Of inserting. placesin this map  si
Prob. II. To project Mercator's. Chart geometrically «... 82
Other methods of projecting Mercator's Chart  Si
Of enlarging ox reducing .maps  . ib.
SECTION VI, — Of the origin and properties of the Rhumb Line,
r or Loxodromic Curve  85
Demonstration, of .tlie increment of the Rhumb being to the
corresponding increment of its latitude in a constant ratio 87
Each point of the mariner's compass expressed by a particular
, . . denomination, by means of the four first points . «i. ... 8*
Tte. method of determining the Rhumb upon which a 1 ship
sails, in .going from one place of the world to another,
exhibited on a Mercator's Partial Map  ib.
The parallel ruler, the readiest. and. most, useful instrument for
/. 1 . . .determining wliatrhomfo thi&is  ....... S9
SECTION Vn. -^Meridional, equatorial, and horizontal construe*-
;ii tionsofmaps  v   , got
Prob. I. Toprojiect the planispifaeres orthographically for a
. map- of .the world, in the position of a right sphere . . . ib.
The fundamental properties of this construction demonstrated
geometrically  -  91
Prob. D„ To find-any- number, of points in the curve of an
«llipse-'d«ft&Ced byje^and through those points to describe.
thatellipse  •  '  93
?bob. IT. To project planispheres orthographically on the
plane of the equator, in the position of a parallel sphere,
lor a -map of the world  ib.

XII TABLE OF CONTENTS^ Page.
Scholium  -....-•  94
Of the stereograpbic construction of maps — first, upon the
plane of a meridian; secondly, upon the plane of the
. equator ; and thirdly, upon the horizon of a place ... 95
Prob. III. To project planispheres for a map of the world
stereographically; on the plane of a meridian, in the posi
tion of a right sphere .  96-
In this- projection the meridians are. drawn from a line of
. ..secants, and the parallels have for their radii the cotau-
. gents of the latitude   ib.
Prob. IV. To project the planispheres stereographically for
a map of the world, on the plane of the equator, in the
.position of a parallel sphere  ........'  97
Of the. stereograpbic projection upon the plane of the. horizon
.ofany.place  ;..<....  98
Prob. V. To construct a map stereographically for a hori
zontal projection of the sphere, answering to the latitude' -
ofLondon,  ..•*  ..-.:....'.-.  ... 100
First,' to, draw the circles of latitude  ib.
Secondly, to draw the circles of. longitude i .  ... — ;.... 102
Prob. VI. The same construction, by . projecting, first, the
meridians; and, secondly, the parallels-of latitude ... 103
The method of completing this niap: - . . .-'s.h .-j .......... 105
Formulae connected with thisprpjection. . . . .  106
Prob. VIL To draw a: horizontal projection with azimuth
lines, to shew the hearing and distance ;of ail places
within the map, from London in the centre .. ..... ...... 107
SECTION VIII. — On tlie principles and practice- of developing
a, spheric surface on a plane  t:. ... Ill
Application of the developemeut of the cenic surface] in the.
. construction of maps   ¦..,  113
Demonstration of the construction of a particular map, that
shall exhibit the superficial. and linear measures in their
truest proportions  ,  ,. n4
An example founded on the foregoing principle, in the con
struction of a map that shall contain that portion of the
earth's surface, situate between 10° and 60° north lati
tude, its longitude being 1 10° from 20" east of the Cana
ries, to the centre of the western hemisphere  J 16

TABLE OF CONTENTS. XlH
. - Page.
The properties which such a map possesses  1 17
Another example of a map, in which the superficies of a given,
zone shall be equal to the zone on the sphere; while, at
the same time, the projection from the centre is strictly
geometrical  ,118
To construct a planisphere on the principles of developement,
for a given sphere or globe  121
Scholium   ,,>  123
"What is meant by developing in a right line all the parallels,
and one of the meridians, that passing through the middle
of the map; whilst the degrees of longitude shall be
subject to the laws of decrease — that is to say, propor
tioned to the cosines of the latitude  124
SECTION IX. — Geometrical projections of chorographical maps
of Europe, Asia, Africa, the Americas, England, Spain,
a part of Italy, the Russian Empire, and a general
method of drawing any partial maps  125
Preliminary observations  127
Four different methods of projecting partial maps ........ ib.
1st. The meridians and parallels being straight lines  ib.
2dly. Both the meridians and parallels being curved lines . . ib.
Zdly. The parallels being curved lines, and the meridians
straight lines  •  •  ib.
ithly. The meridians being curved lines, and the parallels
straight lines  ib.
Prob. I. To project the meridians and parallels for a part
of Italy,. by. the third method   -¦•• 128
Prob. II. To project the meridians and parallels for a map
of, .Spain, by the first method   130
Prob. III. To project a map of Mia by the second method . 132
Prob. IV. To project a map of Ajrica by the second method ib.
Prob. V. To project a skeleton map of Africa by the fourth
method  .-,  '33
Prob. VI. To project a map of North America by the second
method  lb-
Prob. VII. To project a map of South America by the first
method  •  134
Prob. VIII. To project a map of South America by the fourth
method  ib.

Xiv TABLE OF CONTEXTS. **»
Prob. IX. To project a map of Eierope by the second method 134
To construct a diagonal scale for laying down the meridians
according to the law of their decrement shewn in pages
9 and 10  ,3*
Prob. X. To project a map of Europe agreeably to the third
method  *  13&
Construction of a diagonal scale for the meridians, agreeably
to-the foundation ofthe tables of longitude, pages 9 and 10 ib.
Prob. XI. To project a map of Asia by the third method . . 138
Construction of a diagonal scale for the lines of longitude,
according to the law of their decrease, as illustrated in
page 9th, and laid down in page 10th  139
Prob. XI J. To project a- map of England according to me
thod the first  -  l4»
Construction of a diagonal scale for the laying down of the
meridians in this map  •   *»•
Prob. XIII. To project a map of the Russian Empire by
method third  141
Prob. XIV. To project a map of Russia by method second . 142
Prob.- XV. To project a map of any country  ib.
A remark on the term defect, shewing that it only refers to the
common way of considering maps  143
SECTION X. — Of the construction and use of the1 Analemma .144
A definition of it  ib.
The construction of this instrument;- andj ist. To describethe
points of the mariner's compass, the primitive circle
being given ....,  ib.
Qdly. To draw the parallels of the sun's declination  145
3dty. To draw the hour-lines .,  146
Of the index of the A'nalemma  ].J7
Sdli/i Of the use of this instrument  ib.
To find the time of the sun's rising and setting-, with-the length
of the days and- nights ..   ]4s
T» find the beginning and ending of twilight  j£.

TABLE OF CONTENTS. XT
APPENDIX. Pag°'
Containing, 1st. Some methods of drawing large circles and
ellipses; 2dly. Directions for colouring maps ; 3dly. A
catalogue of some of the best maps ; and ithly. General
rules and observations forjudging of the accuracy of dif
ferent maps, when a comparison is to be made between
them, and places inserted from one into another  151
To describe large circles  ib.
1st. By the beam compasses  ib.
2dly. By the ship-carpenter's rule  {]/.
3dly. By the horizontal compasses  ,  152
Athly. By the cyclograph  ib.
Of the centro-linead  153
Methods of drawing an ellipse  ib.
To draw an ellipse upon either the major or minor axis, either
axis being given  154
1st. Upon the major axis  ib.
2dly. Upon the minor axis  155
3dly. By the trammel  fa
To make colours for washing maps  155
Yellow  fa
Blue  fa
Green  ib.
Eed ,  157
Brown  fa
Newman's liquids  fa
Cake colours  ,  fa
Directions for colouring maps  ; . . . 158
To line maps  ib.
A catalogue of the best maps  fa
Planispheres  159
Mercator's Projection  ib.
Maps of Europe  ib.
 of Asia  -. 162
 of Africa  J 63
—  of America  164
General rules and observations for judging of the accuracy of
different maps, when a comparison is to be made between
them and places inserted from one to another  164
Notes and Illustrations  169

A TREATISE

ON THE

CONSTRUCTION OF MAPS.

SECTION I.
PRELIMINARY REMARKS ON THE NATURE OF THE TER
RESTRIAL GLOBE, ITS CIRCLES AND THEIR USES, WITH
THE DIFFERENT POSITIONS OF THE SPHERE.
1. .F OR the purpose of representing more accurately the
globe which we inhabit, geographers have had recourse to
spherical balls, on the surface of which are drawn the various
divisions of the earth. But the relative divisions of the earth,
and the positions of places, cannot be accurately laid down
on these spherical balls, tili certain circles have been describ
ed. And when these balls have been fitted up with such an
apparatus, we are then enabled to illustrate and explain the
phenomena produced by the motions of the earth, and the
diiferent situations of its principal inhabitants. The ball
thus prepared, is called the artificial globe, and what we
have described is properly the terrestrial globe/ so called,
to distinguish it from another of a similar form, and fur
nished in a similar manner, but the surface of which repre
sents the various assemblages of stars or constellations that
B

2
appear in the heavens, and therefore this last is called the
celestial globe*.
2. We have observed, that in order to ascertain the posi
tions of places and countries on the earth, certain circles are
supposed to be drawn on its surface, analogous to those which
are supposed to be drawn in the heavens.
3. The circles of the globe are divided into great and
small; yet they are all commensurate, and supposed to con
tain 360 equal parts, or degrees, each.
4. A degree, or the S60t!i part of the circumference of the
globe, or any one of its circles, contains 60 geographical,
and about 69f statute miles (37.) Each degree is sub
divided into 60 equal parts, called minutes; each minute, into
60 equal parts, called seconds, &c. and more conveniently
marked thus : deg. min. sec.
o / fl
5. Great Circles are such as divide the earth into two
parts ; as, the equator, the ecliptic, &c. (9 and 16.)
For example. If a sphere were cut by a plane into two
equal parts, the section would pass through the centre of the
sphere, and the base of each hemisphere would be a circle,
called a primitive or great circle.
6. The small circles are those which divide the sphere
into parts ; as, the tropics, the polar circles, &c. (41. 43.)
For example. If a sphere were cut by a plane into two
unequal parts, the section would not pass through the centre,
and the base of each frustrum would therefore be called a
secondary or lesser circle.
7. Considering that these circles are really represented'
on the artificial globe and maps of the world, and that ihe
methods of drawing them are the projections contained in

* Encyclopaedia Britannica, 4th Edition, article, Geography^

this work, it will be proper here to consider a Httle more
particularly their nature and uses.
Of the Axis andJ?oles.
8. As the earth turns about on an imaginary axis once in
twenty-four hours, the artificial globe is furnished with a
real axis, formed by a wire passing through the centre; on
this wire the globe revolves. The two extremities of this
axis, are its poles; the one being called the north, and the
other the south pole. And the terms north and south pole,
are equally applied to the extremities of the central meridian
in either hemisphere of a map of the world; the upper
being the north, and the lower the south pole, (49.)
Of the Equator, or Equinoctial Circle*.
9. A great circle, drawn on the globe at an equal distance
from both poles, is the equator, or equinoctial line, and
represents on the globe a similar circle, supposed to be drawn
round the earth, and distinguished by the same names. By
sailors, the equator is usually called the line; and when they
pass over that part of the ocean, where it is imagined to
be drawn, they often make use of various superstitious cere
monies. 10. The equator, like every other great circle, is divided
into 360°; and, when it is bisected in a map of the worldj
by the circumscribing meridian, 180 of those degrees lie in
the eastern, and the remaining 180° lie in the western hemi*

*

I use these terms indifferently throughout this work, though the
first is more strictly applied to the terrestrial globe and geographical
maps; the latter, is properly enough applied by Emerson, to the celes
tial globe, and maps of tlie heavens.
B 2

sphere; and through every 10th or 15th of those degrees,
are drawn the meridians, or circles of longitude, (17.) And
because time is measured by the diurnal motion of the earth,
and this motion is performed on the plane of the equator, the
equator therefore is the measure of time, (39 and 40.)
¦11. The equator being equally distant from both poles,
divides the globe, or map of the world, into two equal parts,
called the northern and southern hemispheres, (22. 45.)
12. Corollary. All places, therefore, that lie between the
equator and the north pole, have north latitude; and those
which lie between the equator and the south pole, have south
latitude. 13, From the equinoctial line, the declination of the sun
or stars is accounted.
14. The equinoctial points are, Aries and Libra, where
the ecliptic crosses the equator ; and when the sun is in either
of these points, the days and nights are equal all over the
earth. 15. The equinoctial line on the earth, passes through the
middle of Africa, in the almost unknown territory of Macoco
and Monemugi, traverses the Indian Ocean, passes through
the islands of Sumatra and Borneo, and the immense expanse
of the Pacific Ocean; then extends over the province of
Quito in South America, to the mouth of the river Amazons.
Of the Ecliptic Circle.
16. The ecliptic is a great circle, cutting the equator
obliquely in the opposite points of Aries and Libra, called
the equinoctial points, for a reason we have already given,
(14.) The ecliptic is designed to represent that circle which
the earth describes in her annual motion round the sun.
This Circle extends on each side of the equator,^ to the lati
tude of 23° §8'; and in consequence of this declination, its
poles are distant from those of the equator, by the same mea-

sure. The ecliptic is divided by astronomers into twelve
equal parts, called signs, each containing 30° ; these twelve
signs correspond with the twelve months of the year.
Of the Meridians, or Circles of Longitude.
17. Through every 15th degree of the equator, there is
drawn on the globe a great circle, which completely invests
the sphere, passing through the poles, and cutting the equa
tor at right angles; but on maps of the world, these circles
become circular arcs, and they are usually drawn through
every 10°. These circles are called meridians, because when
the sun, in his apparent course from east to west, reaches
the corresponding circle in the heavens, it is noon on that
part of the earth over which the meridian is supposed to
pass. Properly speaking, every place on the earth has its
own meridian (18), though, to prevent confusion, these
circles are drawn on the artificial globe only through every
J5° of the equator. Of the "brazen Meridian.
18. To supply the place of other meridians, the globe is
hung in a strong brazen circle, which is called the brazen
meridian, or sometimes, by way of distinction, the Universal
Meridian, or only the meridian. The brazen meridian, like
the equator, is divided into 360* ; but these are marked by
nineties on each quadrant, being on one half of the meridian
numbered from the equator to the poles, and on the other
half, from the poles to the equator,
19. Though the meridians completely invest the globe,
they are commonly, and perhaps properly, called only semi
circles, which is the property of the meridian of any place;
the other half of the same circle, being called the opposite
meridian. There are twenty-four of them in all, allowing

BmOUb **** Sphere

Jarruarorv del .

•finart- dvuOfi.

ing through the centre C, in a direction perpendicular to
the axis P P'. This circle corresponds to the equator, and
it divides the earth of the globe into two hemispheres;
EPQ being the northern, and EP'Q the southern hemi
sphere, (9.)
23. Let G, I, K, represent the situation of three places on
the surface of the globe; through which, let the great circles,
P & P', P IP, and PG F, be drawn, intersecting the equator,
EQ, in n, m, a, respectively. These circles are the meri
dians of the places, K, I, G ; and as every circle is supposed
to be divided into 360°, there must be 90° frqm the equator to
each pole. Hence, the latitude of the place K, is measured
by tlie degrees of the arc intercepted between K and n;
and the latitudes of G and I, are measiired by the degrees of
the arc intercepted between G and a, and I and m, re
spectively. These latitudes will be called northern lati
tudes, because the places lie in the northern hemisphere,
(9 and 22.)
24. Let there be two other places, W and V, in the south
ern hemisphere. The latitude of W, will be measured by
the degrees of the arc intercepted., between W and a ; and
the latitude of V, by the arc intercepted between V and m;
and these will be called south latitudes, (%. 9 and 22.)
The distance between K and n, is sometimes called the dif
ference of latitude ; and so, for the space between I and V, we
should call the extent of that portion of country over which.
I V passed, the difference of latitude.
25. Further, let the circle cedvG, be drawn parallel to
the equator; this circle is called a parallel of latitude, and
as it does not pass through the centre, it is evidently less
than the equator, or it is a small circle. Now all the arcs,
such as R e, a G, &c. intercepted between the parallel and
the equator, must be equal, since the circle is parallel to the
.equator j and hence, every point in this parallel, or -eyery

8
place on the earth through which it is supposed to pass, has
the same latitude.
Latitude is, therefore, the same all over the earth, being
constantly measured from the equator to the poles.
Illustration of Longitude.
26. The longitude of a place is measured by the degrees
of an arc of the equator, intercepted between some par
ticular meridian, and the meridians passing through the place.
Thus, suppose G to represent the particular meridian, and m
to represent the place whose longitude is required ; the lon
gitude of m, is measured by the arc m a of the equator, in.
tercepted between a, and the point where the meridian of G
meets the equator, and m the point of the equator where it is
cut by the meridian of the place m. The particular meri
dian, from which we begin to reckon the degrees of longi
tude, is called the prime or first meridian, and it is different
in different countries *.
Demonstration of particular Longitude.
27. In the diagram referred to above, if G represent the
observatory of Greenwich, a will be the point from which
we begin to reckon the degrees of longitude ; and all places
situated to the east of a, such as R, m, will have east lon
gitude, while those situated to the west, as n, will have west
longitude. The distance in degrees, from a to R, is some
times called the difference of longitude ; and so for the space
R m, we should say, the difference of longitude R m.
28. In reckoning the longitude, we sometimes number
the degrees only as far as 180°, till we come to a again;

* See Note A.

but at other times, they are numbered all round the equator
from the point a; (see Fig i' 7th, Plate X.) for instance,
180°, till we come to a again; hence, in reckoning in the
direction a, R, m, we should say, that everyplace was so
many degrees east longitude; while, if we reckoned in the
direction n E, we should say, that all the places had so many
degrees west longitude all round the equator. To accom
modate the globes to both these modes of reckoning the lon
gitude, the equator is usually divided both ways, in a con
tinued series, from o or zero, at the first meridian, to 360°.
Foundation of Tables of Longitude.
29. It is evident, that as the parallels or circles of latitude
become smaller as they approach the poles, the arcs of these
parallels intercepted between the same two meridians, will
also be smaller as we proceed from the equator to the poles,
though, in fact, they consist of the same absolute number of
degrees. 30. Hence it will be easy to see, that a degree of lon
gitude must be smaller towards the poles, than at the equator;
and must become gradually smaller and smaller, till we arrive
at the poles, where it will vanish, and be equal to nothing.
31. Thus, the arcGu, contains the same number of de
grees as the arc a m, though the former arc is much
smaller than the latter. And between the arc G v and the
pole, the arcs gradually diminish, till they become inca
pable of divisibility.
32. As a degree of longitude is therefore different at every
degree of latitude, it becomes necessary to ascertain the rela
tive proportion between the two; and for this purpose the
following table has been constructed, which shows the abso
lute measure of a degree of longitude, in geographical miles,
and parts of a mile, for every degree of latitude, taking the de
gree of latitude at the equator, equal to 60 geographical miles.

10

TABLE I.

S3. A table showing the length of a degree of longitude
for every degree of latitude, in geographical miles.

Lot.

Geog.
Miles.

Lai.

Geng.
Miles.

Lai.

Geog.
Miles.

Lot. 69

. Geog
Miles. '

1

59,96

24

54,81

47

41,00

21,51

2

59,94

25

54,38

48

40,15

70

20,52

3

59,92

26

54,00

49

39,36

71

19,54

4

59,86

27

53,44

50

38,57

72

18,55

5

59,77

28

53,00

51

37,73

73

17,54

6

59,67

29

52,48

52

37,00

74

16,53

7

59,56

30

51,96

53

36,18

75

15,52

8

59,40

31

51,43

54

35,26

76

14,51

9

59,20

32

50,88

55

34,41

77

13,50

10

59,08

33

50,32

56

33,55

78

12,48

11

58,89

34

49,74

57

32,67

79

11,45

12

58,68

35

49,15

58

-81,79

80

10,42

13

58,46

36

48,54

59

30,90

81

¦ 9,38

14

58,22

37

47,92

60

30,00

82

8,35

15

58,00

38

47,28

61

29,04

83

7,32

16

57,60

30

46,62

62

28,17

84

6,28

17

57,30

40

46,00

63

27,24

85

5,23

18

57,04

41

45,28

64

26,30

86

4,18

19

56,73

42

44,95

65

25,36

87

3,14

20

56,38

43

43,88

66

24,41

88

2,09

21

56,00

44

43,16

07

23,45

89

1,05

22

55,63

45

42,43

68

22,48

90

0,00

23

55,23

46

4i,68

34. As it is often more convenient to estimate degrees of
longitude in English statute miles, we have inserted the fol
lowing table of English statute miles, which will, besides, be
of considerable service in projecting a map of any part of
Britain.
li

TABLE II.

35. Table 2, showing the length of a degree of longitude
for every degree of latitude, in English statute miles.

Lot. 0

Eng.
Miles.

Lai.

¦ -¦ W )...n
Eng:
Miles.

Lot. 61

Eng.
Miles.

69,2000

31-

59,31.62

33,5489

1

69,1896

32

58,6851

62

32,4873

2

69,1578

33

58,0360

63

31,4161

3

69,1052

34

57,3696

64

30,3352

4

69,0312

35

56,6852

65

29,2453

5

68,9363

36

55,9842

66

28,1464

6

68,8208

37

55,2659

67

27,0385

7

68,6845

38

54,5303

68

25,9230

8

68,5267

39

53,7788

69

24,7992

9

68,3481

40

53,0100

70

23,6678

10

68,1489

41

52,2259

71

29,5294

11

67,9288

42

51,4253

72

21,3842

12

67,6880

43

50,6094

73

20,2320

13

67,4264

44

49,7783

74

19,0743

14

67,1448

45

48,9313

75

17,9103

\5

66,8424

46

48,0705

76

16,7409

16

66,5192

47

47,1944

77

15,5665

17

66,1760

48

46,3038

78

14,3874

18

65,8134

49

45,3994

79

13,2041

19

65,4300

50

44,4811

80

12,0166

20

65,0265

51

43,5489

81

10,8250

21

64,6037

52

42,6037

82

9,6306

22

64,1609

53

41,6453

83

8,4334

"23

63,6986

54

40,6751

84

7,2335

24

63,2177

55

39,6917

85

6,0315

25

62,7167

56

38,6959

86

4,8274

26

62,1963

57

37,6891

87

3,6219

27

61,6579

58

36,6705

88

2,4151

28~

61,1001

59

35,6408

89

1,2075

-29

60,5237

60

34,6000

90

0,0060

30

59,9293

1 . ._..

12

Method of reducing Degrees to Miles, and vice versa.
36. Hence it appears, that the degrees of latitude are all
equal (25), and that a degree of longitude at the equator, it
equal to a degree of latitude, as each is Tg^th part of a great
circle. 37. In the second of the foregoing tables, a degree of lon
gitude at the equator, is estimated at 69,2 English miles, or
about 691.
38. The length of a degree in miles, is usually estimated
at 69|, (4), but this is too much. Hence, to reduce degrees
of latitude, and those of longitude near the equator, to Eng
lish miles, it is necessary to multiply them by 69,2, or, if
great accuracy is not required, by 70.
Computation of Longitude in time.
39. As the sun, in bis apparent motion round the earth,
measures a great circle in about twenty-four hours, or, in
one hour, passes over ^th of such circle, or 15°, it is evident
that all places which lie 15° west of any meridian, must
have noon, or any other time of the day, an hour later, than
those situated under that meridian; and that all places
which lie 15° east of any meridian, must have the same times
of the day an hour sooner. Hence, because the meridians
drawn on the globe, make a difference of an hour each in
the time of places, they are sometimes called hour circles-
and the longitude of places is sometimes reckoned in time
as well as in degrees.
40. Degrees of longitude are reduced to hours and minutes,
and vice versa, by allowing an hour for every 15°, and 4' for
every degree. Upon this hypothesis, all machines for mea-
suring time, such as sun-dials, clocks, and watches, are con«
structed.

13

Of the Tropics.
41. Through those two points of the ecliptic, where it is at
the greatest distance from the equator, there are drawn or
the globes and maps of the world, two circles on the former,
parallel to the equator, and on the latter, two circular arcs,
called the tropics. That in the northern hemisphere, is called
the tropic of Canter, as it passes' through the sign Cdncer;
and, for a similar reason, that which is in the southern hemi
sphere, is called the tropic of Capricorn. The two points
through which they are drawn, are called the Solstitial
Points, (44.)
42. The imaginary line, which corresponds to the tropic
of Cancer on the earth, passes from near Mount Atlas on the
western coast of Africa, past Syene in Ethiopia ; thence over
the Red Sea, it passes to Mount Sinai, by Mecca, the city
of Mahomet, across Arabia Felix, to the extremity of Persia,
the East Indies, China, over the Pacific Ocean, to Mexic»
and the island of Cuba. The tropic of Capricorn' takes a
much less interesting course, passing through the country of
Ihe Hottentots, across Brazil, to Paraguay and Peru.
. Of the Polar Circles.
43. If the poles of the ecliptic be supposed to revolve
about the poles of the earth, they will describe two circles,
parallel to the equator, (and 23° 28', if very great accuracy
be required, or) 23|' each distant from the poles of th<;
equator. Two such circles are drawn on the globes, and are
represented by circular arcs on maps of the \iorld, and are
called the polar circles; that in the northern hemisphere,
being called the arctic polar circle, or merely the arctic
circle; while that in the southern hemisphere, is called the
mntarctic polar circle, ox only the antarctic circle.

!4
Both the tropics and polar circles are marked on the
globes and maps by dotted lines, lo distinguish them from
the other parallels, or circular arcs of latitude.
Of the Colures.
44. The meridional circles that pass through the equinoc
tial and solstitial points, are called colures; the former being
called the equinoctial, and the latter the solstitial colure.
The first determines the equinoxes, the second shows the
solstices; and by dividing the ecliptic into four equal parts,
they also designate the four seasons of the" year.
Of the Hemispheres.
45. The hemispheres are those equal sections of the
earth made by the great circles. Thus, the northern and
southern, are separated by the equator, (9 and 12;) and the
eastern and western are divided by the prime meridian (27),
and by the bisection of the equator and axis, these Tiemi-
spheres are subdivided into four quadrants, of 90 degrees
each.
Of the Wooden Horizon.
46. The globes are supported by a wooden frame, ending
above in a broad, flat margin, on which is pasted a paper,
marked with several graduated circles. This broad margin,
is called the wooden horizon, and represents the rational
horizon of the earth, or»the limit between the visible and the
invisible hemispheres. On the paper with which the wooden
horizon is covered, are drawn four concentric circles. The
innermost of these is divided into 360 degrees, divided into
four quadrants. The second circle is marked with the
points of the compass: that is, the four cardinal points,

15
E. S. W. N. and these are again subdivided into eight parts,
or rhumbs. (See Section VI. article rhumb.) The circle
next to that just mentioned, contains the twelve signs of the
zodiac, distinguished by their proper names and characters ;
and each sign is divided into 30 degrees. The last circle
shows the months and days corresponding to each sign.
47. This wooden ring can represent the rational horizon of
anyplace marked on the terrestrial globe only, when that
place is situated in the zenith ; and the method of fringing
the place into this situation, is called rectifying the globe*.
Of the Position of the Sphere.
48. All the possible positions in which you can place the
globe of the earfh-, are reducible to three: 1st, The parallel
tphete; 2ndly, The right sphere ; and 3rdly, The oblique!
sphere. Of the Parallel Sphere.
49. If the globe be so placed, that one of the poles is in
the zenith, and consequently the other in the nadir, it is in
the position of a parallel sphere; so called, because the
equator, E Q,< (Fig. 2nd, Plate I.) coincides with the
horizon; and the parallels of latitude are of course parallel
to it, while all the meridians cut the horizon at right
angles. 50. The inhabitants of a sphere in this position, who are
the inhabitants at the poles, if there can exist such beings,
have the greatest possible latitude. The stars, which are
situated in the hemisphere to which the inhabitants belong,

* See the Note belonging to the Demonstration of the Stereograpbic
Projection, upon the Plane of the Horizon of ariy place, Section VII.

16
sever set, but describe circles all round ; while those stars of
the contrary hemisphere never rise. The sun is above their
horizon for six months, during which it is day; and he is
below their horizon for an equal interval, when it is night.
' Of the right Sphere.
51. If you place a globe in such a position, that any
point of the equator is in the zenith, it is said to be in the
position of a right sphere,, because the equator and its
parallels are vertical, or over the horizon at right angles.
This position is seen at Fig. 1st, Plate II. where the
axis, P S, is in the plane of the horizon, H O, and the equa
tor, EQ,is in a plane perpendicular to it.
52. The inhabitants of such a sphere, who are the inha
bitants of the earth below the line, have no elevation of the
poles, and consequently no latitude. They can see the stars
at both poles. All the stars rise, culminate, and set to them;
and the sun always moves in a curve, at right angles to
their horizon; and he is an equal number of hours above
and below it, making the days and nights always equal, and
producing also two winters and two summers, two springs
and two autumns, in one year.
Of the oblique Sphere.
5®. When either of the poles of the globe is elevated
above the horizon, so as not to be in the zenith, the globe is
said to be in the position of an oblique sphere, in which the
equator and all its parallels, are unequally divided by the
horizon. 54. This is the most common position of the earth, or it
is the situation which it has with respect to all its inhabitants,
except those at the equator (Fig. 1st, Plate II.) and poles.
(Fig. 2nd, Plate I.)

to fizcabaa » •$

£i4to Sphere,

Oblique, J Spher,

Jand&'&n, del

Smart? sculp.

17
55. To the inhabitants of an oblique sphere/the pole of
their hemisphere is elevated above the horizon as many de
grees as are equal to the latitude, and the opposite pole is
depressed as much below the horizon ; so that' the stars only,
at the former, are seen. The sun and all the heavenly bodies,
rise and set obliquely; the seasons are variable, and the days
and nights are unequal. This position of the sphere is re
presented by Fig. 2nd, Plate II. where the equator, E Q,
and the parallels, cut the horizon, H O, obliquely, and the
axis, PS, is inclined to it: hence, this position is called
oblique*.

* See Note B.

18

SECTION II.

PRELIMINARY OBSERVATIONS ON MAPS AND CHARTS, THE
NATURAL DIVISIONS OF THE EARTH, AND THE OBJECTS
THAT DIVERSIFY THE FACE OF A COUNTRY.
56. W E have shown in the preceding section, that the
surface of the earth may be delineated in the most accurate
manner, on the surface of a globe or sphere. This mode of
delineation, however, can be employed only for the purpose
of representing the general form and relative proportions of
the countries on a very confined scale; and it is, besides,
from its bulk and figure, not suited to many of the purposes
of geography. To obviate these inconveniences, recourse
has been had to maps and charts, or delineations of the
earth's surface on a plane; on which the form and boundaries
of the several countries, and the objects most remarkable in
each, whether by sea or land, are represented according to
the rules of perspective, so as to preserve the remembrance
that they are parts of a spherical surface. In this way the
several countries or districts of the earth may be represented
on a larger scale, and delineations of this kind admit of the
most easy reference.
Of the Distinction between Maps and Charts.
57. In maps, the circles of the sphere, and the boundaries
of the countries within them, are drawn as they would ap
pear to an eye situated in some point of the sphere, or at
some given distance above it ; and in maps of any consider
able extent of country, the meridians and parallels of lati-

19.
tude are circular arcs; but if the map represents only a
small district, as a province or, country, those circles become
so large,' that they may, without any considerable error, be
represented by straight lines.
58. In charts, which are also called hydrographical maps,
as they are representations rather of the water than of the
land, the meridians and parallels arc usually represented by
straight lines, crossing each other at right angles, as in the
smaller maps; and in particular parts there are drawn lines,
diverging from several points, in the direction of the points
of the compass, in order to mark the bearings of particular
places. See Plate IV.
59. In maps, the inland face of the country is chiefly re
garded in the delineation ; but in charts, which are designed
for the purposes of navigation, the internal face of the coun
try is left nearly blank, and only the sea coast, with the
principal objects on it, such as churches, light-bouses, bea
cons, &c. are accurately delineated; while particular care is
taken ito mark the rocks, shoals, and quicksands in the sea,
that may endanger the safety of vessels. The depths or
soundings of the principal bays and harbours, and the direc
tion of the winds, where these are stationary or peculiarly
prevalent, are also very exactly marked. See Plate IV.
60. Another distinction is, that in maps, the sea.coast is
shaded, or softened off from the land, towards the sea; whilst
in charts, it is shaded off, or softened from the sea, towards
the inland.
Of the Cardinal Points on Maps.
61. In maps, the upper side represents the north, and the
lower the south ; the right-hand side denotes the east, and
the left-hand side the west. c.2

20

Of numbering the Degrees of Latitude and Longitude
on Maps.
62. All the margins of the map are graduated or numbered;
the upper and lower margins showing the degrees of lon
gitude, and the right and left the degrees of latitude. See
Plates III. and IV". to which you may refer in going over
the following description, and in drawing your maps, as
you will thence be greatly assisted in laying down the
minutim that diversify the face of a country, or the waters
of the ocean.
63. If (he map is on a small scale, only every 10° of lon
gitude and latitude are numbered on the margin; but if the
map is laid down on a large scale, every degree is numbered,
and sometimes every half degree is marked with the number
30 in small figures.
64. The space included between every 10° in small maps,
or between every 2° in those on a larger scale, is usually
divided into ten-spaces, which are alternately left blank, and
marked with parallel lines, to denote the subdivisions of
single degrees or minntes.
65. Through every 10° of latitude is drawn a line, repre
senting a parallel of latitude; and through every 10° of
longitude, or at smaller intervals in each, where the size of
the map will admit of it, there are drawn lines representing
meridians. In some maps these lines are continued from
side to side, or from top to bottom, across both sea and
land; but in other maps they are sometimes only drawn
across the sea. The first meridian, however, and the prin
cipal circles of the sphere, as the equator, tropics, &c. may
always be drawn across the map.

21

Of Latitude and Longitude by inspection.
66. In most maps it is marked whether the longitude is
east or west, and the latitude north or south; but, if this is
not the case, you will easily discover the longitude or lati
tude, by observing towards what part of the map the degrees
increase. If the degrees of latitude increase from the lower
to the upper part of the map, the country delineated lies in
north latitude ; but if they increase from the upper margin
downwards, it lies in south latitude: and if the degrees of
longitude increase towards the right hand, the countries are
in east longitude ; but if they increasetowards the left hand,
the countries are in west longitude.
The method of finding the Latitude and Longitude
on Maps.
67. When the meridians and parallels are straight and
parallel lines, the latitude of a place is found by stretching
a thread over the place, so that it may cut the same degree
of latitude on the right and left side of the map, and that de
gree is the latitude of the place.
68. And to find the longitude, stretch a thread over ,tlie
place, so that it may cut the same degree of longitude on the
top and bottom, and that degree is the longitude of the
place. 69. For instance, if we take the Mercator's Chart, Pf- *e
XI. and stretch a thread over Paris, we shall find that will
cut each side of the map at 48° 50' 14" N. lat. am/ the
top and bottom at 2° 20', E. long. These, therefore, are
the longitude and latitude of that city.
70. On the contrary, if the latitude and longitude of a
place be given, and you are required to find that place,
stretch one thread over the given degree of latitude on each

22
side, and another thread over the given degree of longitude
at top and bottom, and at the intersection of the threads, is
the place required. By this means you may put down in a
map, any place whose latitude and longitude are known.
71. Now, let the meridians and parallels be curve lines.
Then, to find the latitude of a place, a parallel of latitude
must be drawn through it, by the same rules as the other
parallels are drawn, and it cuts the sides at the degree of
latitude at the place.
72. And to find the longitude of a place, draw a circle
of longitude through it, by the same rules as the other meri
dians are drawn, and it cuts the top and bottom at the de
gree of longitude of the place.
73. But as it is troublesome to draw these circles on large
maps, the following method may generally be sufficiently
accurate. 74. To find the latitude, find by a pair of compasses, and
a scale of equal parts, how far the place is from the two
parallels between which it lies, and divide the distance of the
parallels in that proportion, and you get very nearly the
latitude. 75. Suppose, for instance, the distance between the paral
lels to be 5°, and that' one parallel is of 45°, and the other of
50°; and suppose the place to be within 3 parts of the
parallel of 45% and 7 parts of the parallel of 50°; then 5°
must be divided into 10 parts, and 3 of those parts must
be added to 45°, and it eives the latitude.
. This is done by proportion, thus :
W 3X5° 15°
^ X 7, or 10 : 3 : : 5° :  =  . = ,io.
10 10
therefore, the latitude is 46^*, nearly.
77. In the same manner you may find the longitude
nearly. 78. On the contrary, if the latitude and longitude of a

23
place be given, and you are required to find the place, draw
a circle of latitude through the given latitude on each side,
and a circle of longitude through the given longitude at
the top and bottom, and their intersection denotes the place;
or, as you know between what two parallels of latitude and
longitude the place is, you know by what four lines it is
bounded ; and as you know the proportional distance from
each line, you may easily, by trial, find the point.
DIVISIONS OF THE EARTH.
79. By examining a finished map of the world, you will
perceive that the surface of the earth is exceedingly diver
sified; almost every where rising into hills and mountains, or
sinking into valleys ; and plains of any great extent are ex
tremely rare. \ Plains.
80. Among the most extensive plains, are the sandy de-
sarts of Arabia and Africa; the internal part of European
Russia; and a tract of considerable extent in the late kingdom
of Poland, now called Prussian Poland. But the most re
markable extent of level ground, is the vast platform of
Chinese Tartary, in Asia, whidh forms an immense table of
1380 geographical miles, supported by mountains running in
every direction, and is the most elevated tract of level coun
try on the globe. Mountains.
81. The chief mountains of .Europe, are the Doffranal,
Sec. between Sweden and Norway; the Ural Mountains,

* Pinkerton's Geog. p. 368. 3d Ed.

24
between Russia and Asia; the Carpathian Mountains, or
Mount Karpach, between Poland and Hungary ; the Apen
nines, in Italy; the Alps, between Italy and France, Swit
zerland and Germany; the Pyrenees, between France and
Spain. Those of Asia are, the Altayan Mountains, between
Siberia and Tartary ; Mount Caucasus, between Siberia and
Georgia; Mount Taurus, extending from the Archipelago
to the Euphrates ; the Gaur, in Persia; and the Gauts, S. W-
of Hindostan. The mountains of Africa are, the Mountains
of Komri, between Guinea and Komri; the Mountains
Lupata, N. E. of Caffreria ; the Mountains of the Moon, be
tween Ethiopia and Nigritia; and Mount Atlas, between
Barbary and Sahara. And the chief ridges of America
are, the Stony and the Apalachian Mountains in the north;
and the Andes, or Cordilleras, in South America. But, for
a geological account of mountains, and the phenomena of
the earth, I must refer you to Buffon's Epochs of Nature ;
Burnet's Theory of the Earth ; De Lametherie's Theory of
the Earth, a work which abounds in excellent observations;
Herman's Geology ; Dr. James Button's Theory of the
Earth; Dr. Kirman's Theory of Mountains and Volcanoes;
Jameson's Mineralogy ; Werner, Whiston, and Whitehurst 's
Theories of the Earth (98). Oceans.
82. The greatest concavities of the globe, are those which
are occupied by the waters of the sea ; and of these, by far the
largest forms the bed of the Pacific Ocean, which, stretching
from the eastern shores of the coast of New Holland, to the
western coast of America, covers nearly half of the globe.
The concavity next in size and importance, is that which
forms the bed of the Atlantic Ocean, extending between the
New and the Old World's -, and r third concavity is filled by
the Indian Ocean. Smaller collections of water, though

25
still large enough to receive the name of oceans, fill up the
remaining concavities, and take the names of Arctic and
Antarctic Oceans. The Arctic, or Frozen Ocean, on the
north, the Pacific Ocean on the east, the Indian Ocean on
the south, and the Atlantic Ocean on the west, water the
eastern hemisphere. The Arctic, or Frozen Ocean, on the
north, the Atlantic Ocean on the east, the Pacific Ocean on
the west, and the Southern Ocean on the south, wash th«
shores of the western hemisphere. Seas.
83. Smaller collections of water, that communicate freely
with the oceans, are called seas, which sometimes take their
names from the country- near which they flow. Yet there
are large bodies of water, every where surrounded by land,
called seas. Says and Gulfs.
84. A part of the sea running up within the land, so as to
form a hollow, if it be large, is called a bay or gulf; and
these, also, frequently derive their names from one or other
of the countries between which they flow ; but if those por
tions of water be small, creek, road, or haven, is the name
usually assigned them. Straits.
85. When two large bodies of water communicate by a
narrow pass between two adjacent lands, this pass is called a
strait or straits;, and, upon examination, you will find that
they have obtained their names from one or other of the
countries betweeta which any particular current of this de
scription flows.

26

Lakes.
86. A body of water, entirely surrounded by land, is called
lake, loch, or lough; with the exception.of the Caspian Sea,
and the Sea of Aral (101).
87. This term, and its synomies, loch,' or lough, is some
times applied to what is properly a gulf or inlet of the sea ;
as, Loch Fyne, Loch Swilly. Rivers.
88. A considerable stream of water rising inland, and run
ning towards the sea or a lake, is called a river; a smaller
stream of the same kind, is called a rivulet or brook. The
source of a river, is the place or spring from whence it issues
or rises. The mouth or entrance of a river, is the place where
it empties itself into the sea, or into some other river: the
latter is, more properly, the conflux of a river. When the
mouth is very wide, it is called an estuary. The right or
left bank of a river, is that side which is to the right or left
of a person coming from its source to its mouth. The
upper part of a river, is that which is nearest its source ;
and the lower, that nearest its mouth (97).
89. The great extent of land which forms the rest of the
globe, is divided into innumerable bodies, some of which are
very large, but the majority very small. We may divide the
land thus: 1. Continents;^ 2. Peninsulas; 3. Isthmuses;
4. Capes or Promontories ; and 5. Islands.
1. Continents.
90. There are three very extensive tracts of country, which
may all be denominated continents, though only two of

BJemeraary Map skewing ffi& various oty'ecU tfiat diversify Maps.

Hate m \
Joeing page 2.,i i

jiZ-'U&xrn <tet.

27
them have hitherto been distinguished by that appellation.
1. The continents of the Old World, comprise Europe, Asia,
and Africa. 2. The New World comprehends the con
tinents of North and South America: the former, in ex
tent of land, far exceeds Europe; and the latter, is little
inferior to Africa. 3. The other great division forms the
country called New Holland. 2. Peninsulas.
91. A body of land, that is almost surrounded by water, is
called a peninsula. The most considerable peninsulas in
Europe are, Norway and part of Sweden, Spain and Por
tugal, and Italy. In Asia, there are Kamchatka, Further
India, Hindostan, and Arabia. In North America, there are
Greenland, Labrador, and California.
3. Isthmuses.
92. The narrow neck of land, which joins a peninsula to
the main land, or which connects two tracts of country toge
ther, is called an isthmus.

i)

4. Capes or Promontories.
93. A narrow neck of land, stretching far out into the sea,
being united to the main land by an isthmus, is called a
a promontory; and its extremity next the sea, is called a
cape. 5. Islands.
94. A body of land, entirely surrounded by water, is called
an island; and, according to the strict meaning of this defini
tion, the large divisions just mentioned (90), are islands ; for,
it is almost certainly ascertained, that the continent of North

28
America is every where bounded by the sea; and it has
long ceased to be doubtful, that New Holland is in the same
circumstances; and it is generally called the largest island
in the world. But perhaps it would be better to confine the
term to those numberless smaller islands that appear above
the surface of the water.
95. It may assist the young geographer, to compare toge
ther the above divisions of land and water. He will be con
siderably assisted in this, by an examination of Plate III.
I shall only remark, that the large bodies of land, called
continents, correspond to the extensive tracts of water, call
ed oceans, (82 and 90); that islands. are analogous to lakes
(86 and 94) ; peninsulas, to seas or gulfs, (89 and 91) ; isth
muses, to straits, (85 and 92) ; promontories, to creeks, (84
and 93), &c. &c.
Of the Objects which diversify the Face of a Country.
96. The principal objects that diversify the face of a
country delineated in a map, such as rivers, forests, lakes,
roads, cities, towns, forts, &c. are marked in such a manner,
as that they may most easily be distinguished.
97. Rivers. A river is denoted by a black crooked line,
drawn very fine towards the source or head of the river,
and gradually becoming broader as it approaches towards
the mouth ; and small rivers or rivulets, which unite their
waters with those of the principal stream, are denoted by
similar lines appearing to branch off from their confluence
with the first ; and note, that towns are said to be situate
on the right or left bank of a river, when they are to the
right or left of a person coming from its source towards its
mouth. 98. Mountains. Mountains are represented by the figure of
little hills ; and if those figures are placed ina row or line, they
denote a ridge or chain of mountains running across the land.

29
99. Volcano. If the mountain is a volcano, it is denoted
by the appearance of smoke issuing from its summit.
100. Woods and forests. Woods or forests are denoted
by a number of little trees or shrubs placed in a group.
101. Lakes. Lakes are represented by a circumscribed
spot, shaded with dark lines-; and bays or fens by a more
regular spot of the same kind, more lightly shaded, or, where
the map is coloured, painted of a light green.
102. Roads. Roads are represented in a map by two
straight lines drawn parallel to each other, for the principal
roads; or by a single straight line, for the lesser roads.
103. Cities. Cities are denoted by a large house, or the
figure of a church, with the steeple in the middle; and if
the city is the metropolis of the country, it is denoted by a
white circular space in the middle of the house or church. '
104. Small towns. Small towns are usually represented by
circles; and where a small church occurs, with. the gteeple
at one end, it denotes a parish. Where the map is on a
large scale, or represents only a small dsitrict, the towns are
denoted by a group of small houses, or more commonly by
a number of small shaded spots on each side of the road.
105. Fort or castle. A fori or castle, or fortified town, is
denoted by a semi-circular space, surrounded by an angular
edge, representing bastions.
106. Shoals or soundings. The shoals upon the sea-coast
are represented by small dots. The depth of water in bays.
and harbours, by figures denoting the number, of fathoms;
among which is sometimes drawn the figure of an anchor, to
show, that in that place there is good anchorage for ships.
107. Boundaries or limits for countries. The boundaries
or limits that divide countries from each other, are dis
tinguished in maps by dotted lines, drawn round each coun
try or district, in such a _ direction as to show its proper
form. 108. Coloured limits. Where the map is coloured, the

30
countries or districts are distinguished from each other by
the side of the boundary next each being shaded by a dif
ferent colour from that of the adjoining. Thus, in a map
of Europe, the boundary' of France may be shaded green;
that of Spain red; that of Italy yellow; that of Germany
blue, &c*
109. Scale of miles. In one corner of the map there is
usually drawn a scale, divided into a number of equal parts,
by which the number of miles or leagues from one part of
the map to another may be measured. Sometimes the parts
into which the scale is divided, are used to denote geogra
phical miles, of 60 to a degree; but more commonly they
correspond to the miles in use in the country where the map
is made; as in Britain, to British statute miles (35).
110. Fleur-de-lis. To mark more distinctly the bearings
of different parts of the map, there is usually added, in some
blank space, a circle, with four radii, marking the four
cardinal points of the compass ; the north point being dis
tinguished by the figure of a fieur-de-lis, and the east by a
cross +.

* See the Appendix, Article, Directions for colouring Maps.
f Se6 the Appendix, Article, Catalogue of the best Maps: .

¦tftix%$i£..

31

SECTION III.

ON THE PRINCIPLES OF THE ORTHOGRAPHIC, THE STE-
REOGRAPHIC, AND GLOBULAR PROJECTIONS OF, THE
SPHERE.
111. A HE projection of an object, is its representation
upon a plane, and is formed by drawing lines from the eye
to the plane, through every point of the object.
112. And, therefore, according to the position of the eye,
the object, and the plane, the representation will be different.
113. The construction of maps, in general, consists in
making a projection of the surface of the globe, ©n the plane
of one of its great circles, supposing the eye to be placed in
some point of a line drawn from the centre, perpendicular
to that plane.
114. The describing of these projections, depends upon
the principles of perspective, and the projections of the
sphere; but, in the construction of geographical maps, it
is not practicable to give a faithful representation of the
object ; it being impossible to represent truly, a spherical
surface upon^a plane, retaining the true proportion of the
figures, magnitudes, and positions of the countries, with
their relative situation as to the degrees of latitude and lon
gitude. 115. In order to form a just idea of a circular map of the
world, we must suppose the globe to be divided into two
hemispheres, whose bases are to be the circular planes upon
which the surfaces of such hemispheres are to be delineated
(45). But as these spherical surfaces will be larger than
the circular planes, which respectively correspond to them,

32
it is evident, that the map constructed upon those planes,
must be defective (114). This will appear obvious, from
the consideration, that the two diameters which intersect
each other at right angles, for the axis meridian and equa
tor, upon such a map, must each of them represent an arc,
equal to half the circumference of the circumscribing circle,
although, in reality, it may not be equal to two-thirds' of it.
In consequence of this disproportion, the places represented
upon the plane, near the axis meridian and equator, will
be reduced to about two-thirds of the space which they
would occupy upon the spherical surface or hemisphere;
and the arcs, drawn to measure the circles of latitude which
ought to be parallel to each other, are not so in a single in
stance. 116. The projection of the sphere, is a representation of
its circles, poles, &c. upon a plane surface, such as they
would appear to an eye viewing them from some certain
distance, or fixed point (113).
117. The plane surface, upon which the circles are de
scribed, is called the plane of projection, or the primitive
eircle. 118. The place of the eye is the projecting point.
119. The line of measures of any circle of the sphere, is
that diameter of the primitive, produced indefinitely, which
passes through the projected circle:
120. Any right line representing a circle, as 'the equator
and axis meridian, is called a right circld
1®1. In constructing maps of the world, there may be
employed three different kinds of projections, which derive
their names from the position at which the eye is supposed
to view the circles of the sphere.
122. These projections are,
1. The orthographic projection ;
2. The stereographic projection ; and
8. The globular projection.

33
The last may be denominated a projection, though it is not
usually considered so : it is the nearest approximation to the
truth, we have, $nd may certainly claim the name we have
given it.
first, then, we shall treat of the
PRINCIPLES OF THE ORTHOGRAPHIC PROJECTION.
123. If the eye be supposed at an infinitely great distance
from the plane on which the projection is made, all the lines
drawn from their original points on the surface of the sphere,
to the eye, may be considered as parallel ; and, as the eye is
in a straight line drawn from the centre, perpendicular to
the plane of projection, all these lines so drawn, will be per
pendicular to the said plane of projection, and they will
form a cylinder of rays.
124. The projection made by the intersection of the rays
according to this disposition, is called the orthographic pro-
jection, or, simply, the orthography of the original object.
125. In this projection, the parts about the middle of the
map, are pretty well represented; but those towards the
margin, are two much contracted. This projection is,
therefore, so erroneous, that it ought to be entirely rejected
in the construction of geographical maps, though it may be
employed to astronomical purposes, celestial maps, and the
construction of the Analemma. We have, accordingly, in
Section VII. given the construction of an orthographic map,
and in Section X. that of the Analemma.
It will, however, be proper in this projection, to consider
what are the figures of a straight line and of a circle, with
the properties peculiar to each ; and theD, to shew that equal
parts on the surface of the sphere, are not projected into
parts either equal or similar on the plane of projection.

34

PROPOSITION I.
126. The figure of a straight line, A B, is a straight lin§.
in fhe projection. See Fig. A. Plate V.
For, draw A E, B D, perpendicular to xy, the plane of
projection, and join E D, which will represent the intersection
of the plane passing through EAB,D, with the plane of
projection. Draw 7?? n perpendicular to E D, and n is the projection of
m. Thus, it appears, that E D is the projection of A B.
Draw A C para lei to E D ; then will A C D E A be a paral
lelogram, and AC is equal to ED, by Euclid, Prop. 34th,
Book I. AC may, therefore, represent the projection of A B.
127. It will follow from this, that when it is necessary to
make the representation upon a plane, at a distance from the
body, we may suppose the plane to touch the body, as the
parallelism of the plane remains the same. It is obvious,
also, that the axis meridian and equator, are always repre
sented by straight lines, they being great circles of the
sphere, which pass through the projecting point, and the
centre of the primitive. It is evident, too, that the projec
tion of any right line is greatest when that right line is
parallel to the plane of projection ; and, as every line paral
lel to the plane of projection, is projected into a right line
equal to itself, every line that is oblique to the plane of pro-
jection, is projected into one that is less than itself.
PROPOSITION II.
128. The figure of the projection of a circle, is an ellipse,
when the plane of the original circle is inclined to the plane
of projection. See Fig. B. Plate V.
For, let the semi-circle AB C, be imagined to be inclined
to the plane of the paper, which is, in this instance, made

35
the plane of projection; if we draw BE perpendicular to
that plane, and E D, B D, perpendicular to the diameter A C ;
then will E D be the projection of B D (126).
And, by the property of the circle, ADxDC = BDs;
but, as the angle B D E is constant, being the inclination of
the circle to the plane of projection, B D is to D E in a con
stant ratio; namely, that of radius to the cosine of.BDE.
Therefore, D E varies as B D ; and, consequently, D E* varies
as B D* ; hence A D X D C, varies as D E% which is the pro
perty of an ellipse.
The curve, A E Q C, is, therefore, an ellipse. And it was
proved by Prop. I. that the projection will be the same, at
whatever distance the circle is from the plane of projection.
Again, Let O be the centre, and draw O P, O Q, perpendicular to
the diameter A C, then is O Q the minor axis.
Hence, to the radius of the circle, the minor axis is the
cosine of the inclination of the circle to the plane of pro
jection. And, if the circle be parallel to the plane of pro
jection, the projection will be a circle equal to it; that is,
these circles will be identical, being described from the same
centre, and with the same radius,
PROPOSITION in.
129. Equal parts upon the surface of a sphere, are not pro
jected into parts either equator similar, except, as above, in
the case of circles that are parallel to the plane of projection.
By the property of the ellipse and circle, the area ABC ;
the area AEC : : BD : ED : : rad. : cos. BDE.
Hence, by this projection, the area of the circle will be
diminished in the ratio of radius to the cosine of the inclina
tion of the plane of the body to the plane of projection.
And this must necessarily be the case, let the form of the
d 2

36
boly, ABC, considered as a plane, be whatever it may;
because, all ibe lines in the body, are to their corresponding
lines in the projection in that ratio.
It is manifest, also, that the projection is not similar to the
body. Hence, equal inclined circles upon the surface of a
sphere, v. ill not be projected on a plane into circles either
equal or similar. Nor will equal parts upon the surface of
the sphere, be projected into parts either equal or similar by
this projection. PROPOSITION IV.
130. The diameter of a circle, is the projection of a semi'
circumference of that circle, when the plane of that circle is
perpendicular to the plane of projection.
For, if A B C be perpendicular to the plane of projection,
E and. D will coincide, and D is the projection of B. But
the circle ABC, being perpendicular to its diameter AC,
the arc A B is projected into its versed sine A D, and B P is
projected into D O, which is equal to the sine of B P, or the
cosine of A B. And if A B be equal to 60 degrees, then
B P will be equal to 30 degrees, and the line A D is equal to
DO. These two arcs, therefore, one of which is double the
other, have their projections equal. And this must manifestly
be true, both from this Prop, and from Prop. I. ; for to
an eye situated at an infinite distance from P, the whole
circle ABC, would lie in the straight line A C, wliich is
the diameter of the circle *.
We shall have occasion to refer to this Proposition after
wards, in another projection of the sphere; but its connexion
%vith this, rendered necessary the investigation of it here.
¦ -—  - "«g
* See Note C.

37

ON THE STEREOGRAPHIC PROJECTION OF THE SPHERE.
131. In the stereographic projection of the sphere, the
eye is supposed to be situated somewhere upon the surface
of the sphere to be represented, and looking towards the
opposite concave surface. Imagine to yourself, that there
is before you a glass sphere, hung, like the terrestrial globe,
in the position of a right sphere (51); that this glass sphere
has cut upon its circumference, all the lines and circles of the
globe, and that those lines and circles are opaque: that
within this glass sphere, there is a glass plane, coinciding
with the points Aries and Libra, and by consequence, with
the wooden horizon of the frame in which your sphere is
supposed to be hung; then this glass plane meeting the
inner surface, or circumference, of the sphere, would represent
the plane of a great circle of the sphere passing through
Aries, the north pole, Libra, and the south pole. Conceive,
also, that there is a perpendicular drawn from the centre of
this glass plane or circle, to meet the surface of the sphere
precisely where the meridian of 90 degrees is intersected
by the equator, reckoning the degrees lo commence at the
first point of Aries. Suppose, then, that your eye is placed
in this point of the surface, that is, exactly at the extremity
of the aforesaid perpendicular, and that you are looking
towards the opposite hemisphere, on which you would see
all the opaque meridians and parallels, and that straight
lines were drawn from those meridians and parallels respec
tively to your eye, those straight lines would necessarily pass
through the glass plane within the sphere, and on this glass
plane would bevdrawn the true representation of all the lines
upon the surface of the concave hemisphere. This, then,
would be the stereographic projection of the sphere ; and
when you look at -a skeleton planisphere, you have before
you the identical picture that we have been imagining. By

38
turning your glass sphere once round, and viewing the other
concave circumference, you would have another planisphere
depicted on the glass plane, and these would severally repre
sent the eastern and western hemispheres.
But we proceed to investigate the properties of the stereo
graphic projection. PROPOSITION I.
132. The projection of an arc, measured- from the pole,
is equal to the tangent of half that arc. See Fig. 1. Plate
'Kll. for all the propositions under this projection.
Let EQPR be a sphere, the centre of which is O; let
E be the place of the eye, and draw the diameter EQP;
draw also R O Q at right angles to it; and because E is the
place of the eye, QORwill represent the diameter of the
glass plane we have been imagining (131).
Draw any oblique lines E C A, E D B, and join B O.
Now, DO is the projection of the arc BP; but DO is the
tangent of the angle DEO, which, by Euclid, Book III.
Prop. 20th, is equal to half the angle BOP.
Nothing, therefore, can be more evident, than that the
projection of an arc is equal to the tangent of half that arc.
PROPOSITION it.
133. Every oblique circle not passing through the pro
jecting point, is projected into a circle.
For, conceive the chord A B, to be the diameter of a circle
described upon the surface of the sphere, and A EB to repre
sent a cone, having that circle for its base ; if we draw B G
parallel to the diameter Q R, we shall then find that the arc
E Q B, is equal to .the arc ERG.

39
Therefore, the angle EAB, is equal to the angle EBG,
or, as B G is parallel to Q R, equal to the angle E D C.
. Wherefore, as C D, A B, represent two sections of the cone
cutting its sides at the same angle, the sections must be
similar. But the section- A B is a circle, therefore the section
C D is a circle.
Hence, the projection of every circle, is a circle.
PROPOSITION in.
134. The projection of all circles, parallel to the plane of
projection, will be concentric circles.
Join EG, and draw BG, representing the diameter of a
circle, parallel to Q R, the plane of projection ; D H will be
the diameter of the circle of projection, whose centre is O.
Hence the truth of the proposition, that the projection of
all circles parallel to the plane of projection, will be con
centric circles, the radii of which are the tangents of half
the distances of the circles from the pole.
PROPOSITION IV.
135. Every, circle of the sphere, whether great or small,
is projected into a straight line, when the plane of that circle,
or a portion of the same, passes through the eye or projecting
point. The projection of the arc QER, is, by Prop. IV. of the
Orthographic Projection, the straight line Q OR; and the
same for every circle of the sphere passing through E.
PROPOSITION v.
136. The radius of the projection of any great circle, is
the secant of the angle, between the plane of the circle and
the plane of projection.

40
For, produce B O to I, then join E I, and produce E I to
meet QR produced in K; bisect DK in r; and then con
sidering B O I as the diameter of a circle, whose plane is
inclined to the plane of projection in the angle QOB,
D K will represent the diameter of the projection of that
circle. Now D K is equal to O K + O D equal to tangent O E K +
tang. OED == tang. fPI -f tang. fPB = cotangent f
2
P B + tang. |PB = - — ^-fj == 2 cosecant PB=2 secant
^ h 2 sine PB
QB ; therefore | D K *= sec. Q B.
Hence, the radius of the projection of any great circle, is
the secant of the angle between the plane of the circle and the
plane of projection.
From these five propositions it appears, that the projection
of the parts of the sphere will not properly represent in mag
nitude the situation of the parts themselves.
SCHOLIA.
137. If E be the pole of the earth, the meridians, as they
pass through P, will be projected into straight lines, by
Prop. IV. and the parallels to the equator wiH be projected
into circles, whose centre is O, by Prop. III. and if B O I be
the diameter of the ecliptic, D K is its projection, by Prop. V.
This is called the Polar Projection, and was used by Ptolemy*
©f Alexandria, in Egypt.
138. Again, ifEQPR be the equator, then the point on
the sphere vertical to O, will be the pole, and B O I will be
the diameter of any meridian; and if Q be the point from
which the longitude is reckoned, then the projection of the
radius of that meridian, will be the secant of its longitude,
by Prop. V.

* See Note D.

41
PROPOSITION VI.
139. To find the projection of the parallels oflatitudt.
Let E O P represent the plane of the equator, the repre
sentation being a straight line passing through the eye, then
I G is the diameter of a parallel, the projection of which is
HK = OK — OH = tang. | PI — tang. lPG = cot.
|EI — tang, f PG«= cot. 1PG — tang. £PG = 2cot.
PG.
Hence, the radius of the projection of the parallels, is the
cotangent of their latitudes. This is called the Equatorial
Projection, and will be found in the sequel of the work.
The stereographic projection is very convenient for prac
tice, and is very generally employed in constructing maps of
the world, as all the circles are projected into circles or
straight lines, which ate more easily described than any other
figures. OF THE GLOBULAR PROJECTION OF THE SPHERE.
140. If the eye, or point of view, be supposed to be situ
ated at a distance from the sphere, equal to the sine of forty-
five degrees of the circumscribing or primitive circle, the
projection is called the Globular Projection.
141. M.de la Hire* assumed this point of view for this
projection. He produced the axis meridian to a distance
beyond the pole, equal to the sine' of 45° of the plane of
projection; and thence found that the arcs of the sphere,
and their projections, were very nearly proportional to each
other (154).

* Hist, de l'Acsidemie Royal des Scien. 1701, p. 260.

42
Hence, in a map of the world, agreeably to this construe4*
tion, the axis meridian and equator, instead of being divided
, into a lire of semi-tangents, are divided nearly equally, in
like manner as the circumference of the plane of projection.
142. Figure 1st, Plate V. shows how the gradation of
the equator is obtained, when the projection is made upon
the plane of the meridian, the eye at the point X; such as
N X equal to the line s s ; the arc S 5 equal to one half of the
arc ES; whence, C K is one half of CE. For, conceive
your eye at X, looking through E Q, the pellucid plane of
the giass sphere ENQS, and viewing the concave hemi
sphere ESQ, then if s be at the 45th degree of the quadrant
E S, a ray of light passing through s to your. eye at X, would
meet the plane EQ in the point K; and, it is obvious, that.
if C K be equal to K E, rays of light passing from the several
degrees in the arc S Q, through the pellucid plane C Q, to
your eye at X, would divide C Q into ninety, nearly equal
parts, or into nine portions, of ten degrees each (154). ~
143. This is nothing more than placing on the line SN,
the point X, or the line s s, so that the degrees of the equator,
contiguous to the point C, or to the meridian of the middle
of the map, and to the point E, or the plane of projection,
shall occupy the same space on the diameter E Q, which is
easily done by means of the trigonometrical formulae, which
expresses the size of any space, mn.
144. It appears that this projection derived its name from
the conformity it bears to the globe itself. A globular chart
was proposed by Messrs. Senex, Wilson, and Harris*.
This is a meridional projection, in which the parallels are
equidistant circles, having the pole for their common centre,
and the meridians curvelinear and inclined, so as all to meet
in the pole, or common centre of the parallels ; by which

* Mr. Barlow's Math. Diet, Art. Chart.

43
means the several parts of the earth have their proportion of
magnitude, distance, and situation, nearly the same as on the
globe itself, which renders it a good method for geographical
maps. 145. If we assume the point X (Fig. 1st, Plate V.)
beyond the pole N, so that the line N X shall be equal to
three-fourths of the radius C N, we shall come very near the
truth, and the globular projection will then be subjected to
the ibllowing geometrical proposition — " In a given circle to
inscribe any regular polygon," and which may be investi
gated thus : JHOPOSITION.
146. If the diameter DC (see Fig. C. Plate V.) of a
circle, be produced to F, so that the part C F without the
circle, be three-fourths of the radius ED, and if the tangent
DG be drawn, and if DB be _a quadrant, and any point I,
be taken in DB, and the straight line F I be produced to G,
then will the tangent DG be 'nearly equal to the arc DI, and
the equality of distance between any number of spaces on the
radius E B, may also be determined.
'- Demonstration.
Since the part C F without the circle is |ths of the radius,
CF will be |ths of the diameter;
3d Hd
Let D C be denoted by d; then will FD ¦ = d + — = ~;
 ,„„ Ud lld — 8v
andFK = FD-KD = -g-— *> =  g  :
Now the triangles F K I and F D G are similar ;
therefore FK : FD : : KI : DG;
that is, ™=h , ™ : : , : DG - jj^; which is the
value of the tangent :

44
then to shew how nearly this value is to that of the are, we
lid.? *
have

lld—Hv  8b
1 llcf
but s = (dv— -o*)1 = dk v* (I — ~d — g^r &c.)
1  &c.
therefore  L-^tfr&x- *d 8^  =//*,,

l-rr> 1-

llflf lid
( JL£_ 39 g* ^
V1 + 22d + 968d%&c->'
the value of the tangent which should be equal to that of the
arc; but the series expressed in the same terms for the value
' * ( v _l "* o ~\
of the arc, is dP vT x V1 + 6d 40^"' ) 'lt therefore
appears that the value of the tangent is too great to express
the value of the arc.
The two series only agree in the first term. In the series
for the tangent line, the second term is too great, and the
third term much smaller than the third term for the series
of the arc.
When s and v become = — , then, the value of the tangent
lids lid
. ? ,  - becomes -77- which is the same value of the qua
drant as the proportion of Archimedes would give*.
As this series increases nearly equal with the arc itself,
the representation, therefore, of equal portions of all great

* See Note E.

45
circles perpendicular to the plane of projection on the said
plane, will be equal parts very nearly; and their distance
from the centre of the plane of projection, will be
7d*«* (, bv S9r>* a \ c .. , . ,
IT X 1 1 + 22d + 968^, &Cl J *' f°r thC tn^les
FDG, FEO, are similar; therefore, since F D = -£- and
o
EF — ^, it will be, as FD : FE : : DG : EO.
„ Hd 7d
But since —y- is to -^ as 11 is to 7, therefore 11 : 7 : :
a*o* X ( 1 +  + — ¦.  &c. I :  x
V 22 d 968d% J 11
(1 +  r-  ¦ &c. | = EO the distance of the
22 d 968 d* J
projection from the centre of the primitive.
147. Again, suppose we find the point F (Fig. C. Plate V.)
the vertex of an equilateral trangle described upon AB the
diameter of the circle as its base; assume M, a point in the
quadrant A D ; draw M F cutting A B in N j and let E F be
perpendicular to AB ; then the line E F may be found, and
also the distance at which F is beyond C.
Now, because the semi-diameter E A, and the whole dia
meter A B, equal to A F, are given, E F may be had, which is
equal to the square root of AFX — AE*; = (AF1 —
AE1)1. And, because the number of equal parts into which the
diameter is divided for any given polygon, is also given, the
line A N, winch is equal to two of those parts, will be given,
and consequently the part EN may be had. Then, in the
right- angled triangle FEN, there are given the sides EF,
EN, to find the angle EFN; say, as EF : EN : : Rad. :

46
Tang, z EFN. And in the triangle EFM, because the
sides FE, EM, and the angle EFM are given, the angle
E M F may be found thus :
As EM : EF : : sine z EFM : sine Z FME; which,
being found, add to the angle EFM, and subtract their sum
from 180° ; the remainder is the angle FEM; then subtract
ing the angle FEN, which is a right angle, from the angle
FEM, there remains the angle NE'M at the centre.
The part CF may be found thus: the angle E A F = 60',
the angle EAC = 45°; therefore the angle CAF = 15";
and in the triangle C A F, we have the side A F and the side A C
= AE1 + EC1) Tj with *ne angle at A, consequently, having
the two sides C A, A F, and their contained angle, we say,
As the sum of the two given sides,
Is to their difference;
So is the tangent of half the sum of the two unknown angles,
To the tangent of half their difference;
and again, SlnezF : AC : : sine Z A : CF*.
148. Let the distance CF be three-fourths of the radius
E C, and keeping the same construction of the figure, in the
right-angled triangle ELM, EL being equal to ffhs of ED,
we have the two sides, EM, EL, to find the side ML, and
the angles at E and M.
As EM : rad. : : EL : swe^EML, whose complement
is the angle L EM; now, to find LM rad. : E M : : cos. z
EMLsMLj or

LM = ME1 -EL^. And if EM = 1, EL = -75;
therefore, if L M = V ME1 — EL1) then will
VME* — EL*= V lz — 75\
Again, in the right-angled triangle F L M, we have given

* See Note F.

4T
the sides FL, ML, to find the hypothenuse FM, and the
angles at F and M.
To find the angles,
M L : LF : : rad. : tang, z L M F, or cos. z M FL.
To find the hypoth.
Cos. zLMF : rad. : : ML : MF
"Now, if we take the angle LME from the angle LMF,
there will remain the angle E M F; and if we take the angle
MEL from NEL, there will remain the angle NEM;
wherefore, in the triangle ENM, we have all the angles
given, together with the side ME; and the line EN may
thence be found thus :
Sine. Z N : M E : : sine z M : to opp. side N E.
And in this manner may each of the nine spaces in E A be
computed. 149. Now, if we compare these two last articles by the
one which precedes them, we have
AF*— AE*=EF*; and if AF = 2, AE= 1; then
21— ¥= 4—1=3 = EF, whose square root 1-73205 is the
line EF. But by the latter, since CF = |ths of 1 or • 75,
therefore EF= 1-75; wherefore, the difference is -01795;
and by this difference does the apex, F, of the equilateral
triangle, fall within F in the line EF, when C F is = fths of
the radius.
150. Upon the whole, then, it appears, that La Hire's me
thod is as good as either of the other two. That in wliich
we assume the point of view, distant from the sphere three-
fourths of its radius, will not create any material error in the
whole; and that in which the point of view is fixed on the
vertex of an equilateral triangle, having the diameter for
its base, differs from the latter method so little, as not to be
perceptible in the whole projection. Either of them may
therefore be used at pleasure, or La Hire's may be chosen
in preference to both .

48

On the disproportion of the Distances between the Meridians
and Parallels in the Stereographic Projection.
151. In the stereographic projection, the meridians are at
Unequal distances, the greater being at the primitive meri
dian, and decreasing as they approach the centre. There is
also a similar inequality in the distances of the parallels of
latitude being greatest at the poles, and decreasing as they
approach the centre. Equal spaces on the globe of the
earth, are thence represented by unequal spaces on the plana
map. 152. By inspecting Fig. 3d, Plate VI. you will discover
this inequality. In the semi-diameter CQ, the space C 80
is very nearly equal to the half of S 80 on the primitive
circle; and by consequence, 10 Q on the equator, is nearly
equal to Q 10 on the primitive circle. If, then, on the semi-
tangetical line C Q, the first ten degrees towards the centre of
the map, are only nearly equal to one half of their cor
responding ten degrees on the primitive circle; but the first
ten degrees on the same semi-tangetical line, towards the
primitive circle, be nearly equal to ten degrees on the primi
tive circle, as must be the case from the semi-diameter, or
line of semi-tangents being half way between that and the
real tangents ; then equal spaces on the globe of the earth,
are represented by very unequal spaces on the plane map.
153. But to investigate this yet more accurately, because
the surface of a sphere is equal to four times the area of the
circle, whose diameter is that of the sphere, the whole sur
face of a sphere is to one of its great circles as four to one;
therefore, when the eye is on the1 surface of the sphere, the
ratio will be that of two to one. The parts at the centre of
the map are, therefore, only one-fourth part of their corre
spondent originals; but those at the extremity are nearly
equal, and would be exactly so, taking* the ultimate ratios j

49
and, consequently, the whole map cannot be a just ratio of
the spheric surface.
On the equality of the Distances between the Meridians
and Parallels in the Globular Projection.
154^ In the globular projection, the meridians and paral
lels are equidistant, when measured on the equator and axis
meridian ; and, therefore, equal spaces on the spheric surface*
are represented by equal spaces on the map, by an approxi
mation as near the truth as any projection will admit ; for
a spherical surface can, by no possible method, be represented
exactly upon a plane.
155. When the eye is posited in the surface of the sphere,
as, is the case in the stereographic projection, all the meridians
and parallels are represented on the plane of projection as they
really are on the surface of the globe itself — they are all cir
cular; but when the eye is placed out of the surface, the meri
dians and parallels depart from the form they possess on the
terrestrial globe — they cease to be circles, and become ellipses,
whose eccentricity is in proportion to the distance of the eye
from the surface of the sphere. The meridians and parallels
in the globular projection are, therefore, ellipses. They are
not usually drawn so, geographers contenting themselves with
circular, in place of elliptical curves; and, as the deviation is
excessively small in this projection, we have followed the po
pular mode. Mr. Peter Nicholson, who places the point of
view in the globular projection at f ths of the radius (145), has
given the real elliptical curve of the meridians and parallels *.
To a skilful geographer, the real method of projection must
always be the smallest of the difficulties presented in the exe
cution of the map, especially as there are many simple and

* See Dr. Rees's Encyclopaedia, Art. Proj. of the Sphere, Prob. XI.
Meth. 2. Ex. 1, 2, and 3. Plate IX. Fig. 1, 2, and 3.

50
convenient methods of drawing ellipses. The horizontal prd-
jection performed after Nicholson's principles*, would be capa
ble of giving distances, as well as the stereographic ; for, there
is no property of the stereographic projection, that can recom
pense in planispheres, the inconveniences of the disproportion
thence arising between the spaces that ought to be equal.
156. The orthographic projection has, with regard to
spaces, the contrary defect from the stereographic, as it
diminishes them from the centre to the circumference, on
account of the obliquity under which the lateral parts of the
sphere are represented to its diametral plan; but when we
produce the optical axis out of the sphere (140), the plane
of the picture still passing by the centre, we find that there
exists on the produced axis a point (141), where the in
equality of spaces is the smallest possible (142); for, it is
evident, that when the point of view is at such a distance,
that the obliquity of the rays, which tends to enlarge the
spaces, becoming smaller, may be compensated by that of
the projected surfaces, which fends to diminish them, and
their increase must be changed into decrease +. There cannot
be absolute equality in all, because the law of their variation
depends upon their particular situation; but at the limit
Which we have assigned, their differences are sufficiently
small to be neglected in a general map. This will be suf
ficient^ evident, by comparing Fig.' 1st, Plate VI. with
Fig. 1st, Plate V. In the former, the lines drawn from N,
to the equal divisions of the quadrant QS, have more ob
liquity than the lines drawn from X to the equal divisions
of the quadrant Q S, in the latter; and hence the divisions
in CQ, Fig. 1st, Plate V. become nearly equal among
themselves; whereas, the divisions on CQ, in Fig. 1st,
Plate VI. are very unequal, being greatest towards Q.

* See Proj. of the Sphere, by the rules of perspective, in Dr. Rees's
Encyclopedia. t La Croix's Math, and Crit. Geo.

51
Of the two last methods of projection, the only advan
tage the stereographic has over the globular, is, that the
parallels of latitude and the meridians, intersect each other
at right angles ; the globular is equal to the stereographic
in point of facility, and much superior to it in point of cor
rectness, for planispheres or particular maps.
Illustration of a Planisphere.
157. In constructing a map of the world, as well as in
most partial maps, the part of the sphere to be represented,
is supposed to be in the position of a right sphere (47). In
this mode of projection, the hemisphere to be represented,
is supposed to be delineated on the plane of that meridian
by which it is bounded, in the same manner as its concave
surface, conceiving the sphere to be transparent, would
appear to an eye placed in the opposite hemisphere, where
the equator crosses a meridian ; that is 90° distant from that
which forms the plane of projection (131). In a picture of
this kind, we have seen that the meridians and parallels of
latitude, are represented by arcs of circles, except the equator
and axis meridian, which are straight lines ; and each paral
lel or meridian, forms an arc of a great circle, in proportion
as it approaches nearer to the centre of the map. Moreover,
the projection of a map of the world, in the position of a right
sphere, on the plane of a meridian, produces two hemispheres;
the one the eastern, and the other the western hemisphere.
158. The projection of a map of the world, on the plane
of the equator, in the situation of a parallel sphere (49),
produces the representations of the northern and southern
hemispheres, which appear as their convex surface would be
seen by an eye placed at the opposite pole. In this way the
meridians become straight lines, diverging from the same
centre, and the parallels are all concentric circles ; and" equal
spaces on the earth, are represented by equal spaces on the
map, as nearly as in the former cases.

m

SECTION IV.

CONTAINING GEOMETRICAL AND TRIGONOMETRICAL PRO
JECTIONS OF THE SPHERE, ON THE PLANE OF A MERI
DIAN AND THE EQUATOR*. PROBLEM I.
159. To draw the circles of longitude geometrically, for a
map of the world, according to the globular projection of the
sphere, on the plane of a meridian. See Fig. 1st, Plate V.
160. Draw the straight line E Q, of any convenient length,
divide it into two equal parts in the point C, and with the
radius CE or C Q, describe a circle : then, through C, draw
NCS, at right angles to E Q. The lines EQ,NS, repre
sent two great circles of the sphere (120): the former, the
equator ; the latter, the central meridian of the hemisphere
in which it is placed ; and N, S, are the poles (8).
161. Divide the quadrant of the circle QS, into nine equal
parts, 10, SO, 30, &c. and bisect the quadrant ES, in the
point s; this point * is situated in the 45° of the quadrant :
then, through the point s, draw the straight line ss, perpen
dicular to C S, or parallel to E C, and * 5 will be the sine of
45° (142).
i' 162. Produce the axis meridian beyond the pole to X,
and set off NX equal to the line ss (142); then, from the
point X, to the sub-divisions in the quadrant QS, draw'the
oblique lines X 10, X 20, &c. which will divide the semi-
diameter CQ, into nine nearly equal parts (146). The same

* See Note G.

5>3\
operation being performed in the quadrant ES, will divide
the radius C E into nine nearly equal parts. But, it, is qb vious
that all the foregoing work is unnecessary, as f^e.quatoi| mayr
be very easily divided into nine equal parts, ftyi^hputit.
The exhibition of it, however, .gives a connect, idea of,. $e.
manner in which the gradatipn,of the equator js, obtained;,, j,^.
163. Secondly. Through the points, of .these equa^ sub
divisions in the equator, the circles (155), <jf longitude are
to be described from centres found in the following manner.
See Plate V- Fig. 2nd.
From the poles N and S, draw straight lines to Abe. divi
sions 10, 20, &c. in the equator ; divide the radii CN,;CS,
into two equal parts in the points T, T, and draw T P, T P,
perpendicular to the axis NS; these lines will also. jb-q
parallel to the equator, and they will divide into two equal
parts, all the lines drawn obliquely from the., poles to the
points of sub-division 10, 20, &c. injthe radius OQ;; as,is,
evident from what was shown, Prop. IV. Orthographic PrsQ*
jection, that the projection of the arc NP, (=.60°,) isequal
to the projection of the arc PQ, (=30°.), , . ,; ^
164. Produce QE towards R, and from the points of
intersection formed by the lines TP, TP, crossing the lines
drawn obliquely from the poles N and S, tq. the points 10,
20, &c. in the radius C Q, raise perpendiculars cutjingjeajch
other in the equator produced towards R. The points ...at,
which these perpendiculars cross each other in the produced
equator, determine the centres of the eight circular arcs of
longitude, for the semi-circle NQS. And the centres for
the meridians in the remaining semi-circle, are found in the
same manner, by only producing E Q, and letting fall upon
this line of measures perpendiculars, as in the former case.
165. The foregoing process is briefly this : (See Fig. 3rd,
Plate V.): — The lines EQ and NS, are at right angles,
and have already been defined — the radius CQ, is divided
into nine equal parts — CN and CS are bisected in the points
P,P— the lines PT, PT, are drawn perpendicular to the axis

54
NS-from N and S, to the point 30 in CQ, are drawn the
straight lines N 30, S 30— the horizontal and parallel lines
PT, PT, bisect N30, S30, in the points oo— from these
points (o, 6) are raised perpendiculars o v, ov, crossing each
Other in v, or the 130th sub-division of the equator, this
point of intersection v, is the centre of the circle of longitude
matked 30, and if one foot of the compasses be placed im-
rrioveably in this point v, and the other extended to the pole
N or S, or to the point 30, it will describe the circular arc
N 30 S. When the plane of projection is so large, that
ordinary compasses will not describe the circles or ellipses of
longitude and latitude, by consulting the Appendix, you will
be enabled to adopt such means and instruments, as to ef
fectually answer your purpose *.
166. The remaining circles of longitude are described ki
the same manner, as will appear sufficiently evident by in
specting Fig. 2nd, which shows at one view every line for
determining the centres of the circular arcs of longitude in
the semi-circle N Q S.
167. The drawing of these circular arcs, is nothing more
than describing a circle through three given points; but we
shall give another method under Problem III. and either
may be adopted at pleasure.
168. It will save much time and labour, if, when yOur com
passes are extended to the radius of the circular arc N 10 S,
the other three circles of longitude of the same magnitude be
drawn before the instrument is shifted for the radius of 20*;
and when extended to the radius of 20", sweep all the circles
of 20° from the circumference of both hemispheres. You
may pursue the same plan with all the other meridians, by
setting off the distance of each centre, as these centres appear
in Fig. 2nd, from C in the diameters of both hemispheres
produced the contrary way.

* See Appendix, article, Method ofDratving large Circles or Ellipses,
on Maps.

55
169. We shall herp give a very beautiful problem, by which
we are taught to describe a circle through any point, and wh ich
will therefore resolve the projection of a circle through three
points, and be of great service to you in describing any cir
cular arcs that occur in this work, or in any practical case.

PROBLEM A.
170. To describe a circle through any point D, in the
semi'rirck A C B. See |Fig. 4th. Plate VI.
171 . Let A C B be a semi-circle, of which A B is the dia
meter, and OC, perpendicular to AB, a radius, and D the
given point. Draw through D a straight line ADE, meeting the
.circumference in E ; join O E, and draw E F at right z * to
O E, and produce OC to F; then, by Euclid, 15 Hi Prop.
Book I. the z CDE == Z ADO, and /DAO is the com
plement of the zODA, ,the ^ A O D being a right z ¦ But
the z O E F is a right angle, and it is equal to the two z s
O E D, D E F ; therefore, the angles O E D, D E F are equal
tothe^sODA, DAO; take away the z O E D = OAD
(by Euc. V. 1. B.), and the remaining z DEF = remaining
,^ODA = ,<£FDE. Wherefore, FDE is an isoceles tri
angle, the vertex of which is F; and hence, with F as a
centre, a cjrcJe;may be described, that. shall pass through the
point D,, which was to be done. And this is the true doc
trine of, describing a circle through. one, or even, three points,
in the projection of the sphere.
PROBLEM II.
172. To draw the circles of latitude geometrically for a
map of the world, according to the globular projection of the
sphere on the plane of the meridian. See Fig. 3d, Plate V.

56
173. Keeping the construction already made use of, pro
duce the equator both ways to the points X, X, and make
EX, QX, each equal to ss, the sine of 45° in Fig. 1st.
Then, to the divisions in the semi-circle ESQ, draw the
oblique lines X 10, X20, &c. both ways, and they will
divide C S, the semi-axis, into nine nearly equal parts. And
through these points in CS, we have to describe the circles of
latitude, from centres found in the following manner :
174. Secondly. Suppose we were required to draw the
circle representing 60° of south latitude, we have here three
points, 60inES, 60inCS, and60inQS, through which
that circular parallel must pass; and if we bisect the two
oblique lines 60 60, 60 60, which meet in C S, we shall have
the points a and c, from which to raise the perpendiculars
ab,cb; b is the point, about which, with the radius b 60,
you are to describe the circular arc, that will represent the
parallel of 60° south latitude.
175. The other parallels are drawn in a similar manner,
from centres lying in the line of measures CS, produced be
yond S ; and if the same process be employed on the other
semi-circle E N Q, all the parallels for one planisphere will
be completed. The tropics and polar circles are laid down
in the same manner, by setting off 23f from the equator and
poles. We may observe here, also, that when you have got
the radius for one parallel, 60? for instance, all the others of
this magnitude may be described, and so on for the remain
ing ones, describing all the circles of the same radii before
you alter the instrument; or, if you divide CS into nine
equal parts, an arc of->eaeh ¦parallel may be described by
Problem A.
176. The other method to which we alluded (167), was
employed by Mr. Arrowsmith, in constructing his large map
of the world. It requires, however, considerable address, to
draw the individual degrees by this or any other method ; but
even this is practicable to the greatest nicety, where there

57
are patience and assiduity. Problems III and IV. shew this
method*. PROBLEM III.
177. By the globular projection of the sphere, and on the
plane of the meridian, to project geometrically the meridians
for a map of the world. See Fig. 1st, Plate VIII.
178. About the centre C, with any radius, as CB, describe
a circle representing the plane Of projection; draw the
diameters N S and A B, crossing each other at right angles ;
the former of these will represent the axis meridian, and the
latter the equator. Then divide each semi-diameter into
nine equal parts, as seen in Fig. 1st; and each quadrant also
into nine, as shown in Fig. 2nd, Plate VIII. thus will the
diameters be divided into eighteen equal parts each, and the
circumference into thirty-six, which may again be sub
divided into 180° and 360°, by the mere subdivision of the
great divisions into ten equal parts each.
179. The next object is to find thie centres from which to
describe the meridians passing through every tenth degree of
the equator.
180. Suppose we are to draw the meridians of 80" and 50"
west of Greenwich, \ye have here three points given, the two
poles, and the points 80° or 50° on the equator, and a circle
is to be described, that shall pass through these three given
points ; an arc of this circle will be the meridian of 80° west
longitude. -, ...
181. About the point S, as a centre, with the radius, SC,
describe a circular, are, as lha,t,dptfed by X C X ; with the
same radius, and about N as centre, . describe the dotted arc
ZCZ; then about the centre 80?,, with the same radius,
describe the arcs, 1, 1; 8,2, crossing the former ; draw
... . \
* See' Mr. Arrowsmth's Comp. to his M*ap off he World. 4to. Lond. 1794.

58
straight lines from 2 to 1, one on each side of A B produced
in D ; — D is the centre of the circular arc, representing the
meridian of 80° west of Greenwich ; and, with the same
radius, the meridians 140* west, 40» and 100° east longitude,
may be drawn.
182. The 50th degree of longitude west, has for its primi
tive centre the point 50 in C B, from which the arcs a, a;
b, b, are drawn crossing the dotted arcs Z, X : on either side
of A B, are drawn the straight lines b a E, b a E ; E is there
fore the centre of the meridian of 50' west longitude, and the
same radius will describe the meridians 170° west-, 10* and
130° east longitude. And you can draw all the other meri
dians in a similar manner,' by using N as one point, the
required degree as the secoad, and S as the third, through
which the circle must <pass>.

PRpBLEM IV.
183. To draw the parallels of latitude according to the
method of Problem III. on the plane of a meridian, for a
map of the world. See Fig. 2nd, Plate VIII.
184. Keeping the consttuction of the foregoing: problem,
of which the plane of projection is identical to Fig. 2nd, we
have in this last diagram, the axis meridian and circum
ference of the plane of projection graduated, or divided, the
former into 18, and the latter into 36 equal parts; and
suppose we were to draw the parallels of 30° and 60* north
latitude, we should proceed thus i •
185. First. For 30% about the 30th degree, with M as
a centre, and with any radius, describe the circle L K T, and
about the points 30% 30' in the primitive circle, with the
same radius, you describe the arcs s,s; n,n cutting the cir
cle LKT; then, through the points of intersection, draw
straight lines, and the point where they meet in the line of

IAGEAMS
^EidraiilbMMg1 tfo<&
Globular ^ro/ectwn ofth&JlferufazA<s
Parallel?

page 46.

'4 W N

Fig<2.

59
measures NS produced, as in R, is the centre of the arc,
that will represent the circular parallel of 30' north latitude.
186. Secondly. For 60% about the centre O, with any
radius, describe the circle PG H, and about the points 60%
60°, as centres, in the primitive circle, with the same radius,
describe arcs c, c; d, d cutting the circle P G H ; then,
through the points of intersection, draw straight lines, and
the point at which they meet, in N S produced, as in J, is
the centre of the arc, that will represent the circular parallel
of 60# north latitude *.
187. The other parallels are drawn in a similar manner,
observing, however, that the first circle, such as LKT, must
have for its centre that point in the central meridian, through
which the parallel is to be drawn. This projection is fully
exhibited on Plate XVI I. where all the meridians, parallels,
tropics, and polar circles, ¦are completed.
188. If the map is very large, and the paper on which it
is to be drawn, will not admit of so many circles, the centres
of the meridians and 'parallels are more easily found in the
following manner.
189. Having divided the semi-diameters and q" >dran£s
each into nine equal parts, find from a scale of equal j.orls,
the length of half the chord of each arc," and the versed sue
of half the same arc ; then add together the square of half
the chord, and the square of the versed sine, and divide the
sum by the versed sine ; the quotient is equal to the diameter,
and i of this to the radius of the circle required. In this
manner the radii of the meridians and parallels may be
found +.
190. The two following Problems exhibit in their contents,
trigonometrical methods of the globular projection on the
plane of a meridian.

* See Note H.
t Encyclop. Britan. Vol. IX. Part II. Page 541. TTfth Edition.

60

PROBLEM V.

191. To project the circles of longitude for a map of the
world, on the plane of the meridian, according to the globu
lar projection of the sphere, from a trigonometrical table of
radii. See Fig. 3rd, Plate VIII.
192. Draw the circle W N E S, as the plane of projection,
and draw WE, N S, at right angles to each other, the
former will be the equator, and the latter the axis meridian.
Then divide each arc of the quadrant into nine equal parts,
and the two diameters also into nine equal parts.
193. The meridians, as we have already seen, must pass
through three given points, the poles, and a division on the
semi-diameter, as N oS, or N bS, &c. The centres for these
arcs will be in the line of measures CE produced; and the
centres for those on the other side, will be on C W produced.
194. Observe, that the radius CW, is divided into 90
equal parts ; and the following table shews the length of the
radius of each meridional arc in those parts.
Trigonometrical Table of Radii for the Meridians.

For the arc SaN the radius a a — 90-61 say 90$ ~) -g
 S&N 

 ScN
— SdN
 SeN
— S/N
— SgN
— SAN

bb

=

92-82

93

cc

=

97-32

¦ —

97k

dd

=

106



106

ee

==

121 1



121

ff

=

149-7



149|

gg

=

215-6



215f

hh

=

410-7



410|J

_ ©
- o>
t II

' '3

CO
r *

2
« w
en .a
© -»*
O O

And with the radius a a, i. c. a 90 j, you can describe the arc
S a N, and so on for any of the others.

61
195. The following Problem shews the canon of radii we
we have given for this method of projecting the sphere on the
plane of a meridian.
PROBLEM B.
See Fig. 4th, Plate VIII.
196. Let N, S represent the two poles, and W E the equa
tor, it is required to find the radius of an arc that will pass
through the three given points N aS.
197. In the right angled triangle N a C, the base and per
pendicular are known, and of course the angle C a N, and
the hypothenuse Na, by trigonometry thus:
a C : C N : : Rad. '. tang, z a, or cot. z N ; and
cos. a z '• Rad. : : a C : a N, or
Rad. : Sec. Za : : a C : a N.
198. Bisect N a in D, and if a perpendicular to N a, be
drawn from D, it will intersect WE, and form a right-
angled triangle, of which the base D a, and the angle D a B,
are known, and of course the hypothenuse a B, (as above,)
which is the radius for describing the arc N a S ; and so oft
for the other arcs N 6 S, &c.
199. There is another way of dividing the radius, as for
example, into a hundred equal parts. See Fig. 1st and
2d, Plate V. in which we might, by any convenient scale
of equal parts, divide C E or C Q, into a hundred, and then
the circular arc answering to

10° -will be equal to  12
20°'-  ¦  ¦  .'  1 25
30°   42
40°  •  ... 61-5
50°,   88
6oo   133
70°   208
80"  ——   ^...,~.. 464

Of those parts set off from
C towards Q or E, added
to the distance between
C, and the several points
10, 20, &c. in the radius
C E, or C Q.

Thus the radius of the circle of 10* longitude, fe equal to th«
dip'^.f-re between QC, and 10 in the line CE; the radius of
that o( 50°, is equal to the distance between 50 and 50; and
that of 80°, between 80 in the line Q C, and a given point
in 0 E produced, which, taken from C, will be equal to
342 2 parts of which radius is equal to a hundred*.
PROBLEM VI.
200. To draw the parallels of latitude from a trigbWh
metrical table of radii, for the globular projection of the
sphere on the plane of a meridian. SeeFig. 4th, Plate VIII.
201. Retaining the primary construction of the foregoing
problem, we have here also three points, through which the
parallels are to be drawn ; namely, one of the divisions 1c, I,
or m, &c. upon the axis meridian NS, and the two corre
sponding divisions of the primitive circle NESW. The
centres for these arcs, will fall on the line of measures NS
produced both ways , and the following table shows the
length of the radius of eapb , in equatorial parts, as in the
former case in meridional arcs.
202. Trigonometrical Table of Radii, for the Parallels of
Latitude.

For the arc 10A 10 the radius
— ti  20 1 20 
 23*. T 23*,  T.
 30 m 30 
¦  .40 n 40 
 50 o 50 . 
 60 p 60 
 66J A 66|  A.
 70 q 70 
 80 r 80 

Ick = 703.5

say

703£t

// = 337-5

—

337|

Tropic= 281-4

—

28l£

mm = 210

—

2J0

n n =143

—

J 43

o o — 97-71

—

97|

p p = 65-3
Arctic = 48-19

—

65£

—

48-f

qq = 39-75

—

39J

rr —. 18-44

—

18*

<° _•
o o
•3 oi
til O •—
ta u
V.-S.5 ilo

* Goldsmith'* Poplar geography, p. 578. Sixth Edition.
+ In tbe diagram, the point 703A h goes beyond the plate, though »xpre< sed
¦pon it.

63
203. The following Problem shews the canon of radii we
have given for drawing the circles of latitude in this method
of projecting the sphere on the plane of a meridian
PROBLEM C.
See Fig. 4th, Plate VIII.
204. Let N, S represent the poles, W E the equator, it is
required to find the radius of an arc that will pass through
the three points 30 m 30, in the quadrant W N E.
205. Draw 30 F perpendicular to CN, and 30 F will be
known, and E C, as sine and cosine of 30° ; whence C F —
C m gives F m, one side of the right-angled triangle 30 F m ;
consequently, in the right-angled triangle 30 F m, having
the z at F, and the sides 30 F, Fm given, the side 30 m,
and the angle 30 m F may be found thus, by plane trigono
metry: As m F : F 30 : : Rad. : tang z m, or, cot. z F 30 m;
and cos. Z m : Rad. : : m F : m 30, or
Rad. : sec. Z m. : : m F : m 30.
206. Bisect 30 m in G, and if a perpendicular be drawn
to 30 m from G, it will intersect C N, or CN produced, and
form a right-angled triangle, of which the base G m, and the
angle GmF, are known; and, consequently, the hypothe
nuse, which is the radius for the arc 30 m 30, may be deter
mined from the above formula ; and so on for the other arcs
or circles of latitude, 10 k 10, 20 1 20, 40 n 40, &c.
PROBLEM VII.
207. To project a map of the world geometrically on the
plane iof '(he equat'dr^ according to the globular projection of
the sphere. See Plate IX.

64
208. About C as a centre, and with any radius, as C B, de
scribe a circle which will represent the equator, as A E B Q.
Draw A B and E Q at right angles, and there will be formed
the quadrants AE, EB, BQ, QA, each of which you
divide into nine equal parts, representing 10° each ; and if
you subdivide each of these tenth parts again into ten equal
parts, the whole circumference will be divided into 360*.
See article 178.
209. Now, to draw the meridians, which, in this case,
will be right circles (120), or straight lines (158); from the
centre C to the divisions in the plane of projection, draw
the straight lines C 10, C20, &c. representing the meridians..
210. To draw the circles of latitude, which, in this in*
stance, are concentric, divide any meridional line, as CE,
into nine equal parts; and through each of those points, from
C as a centre, describe the concentric circles 10, 20, &c. which
will represent the circular parallels of latitude. Draw the
other hemisphere in the same manner, and you may then
fill up your skeleton map with all the variety of objects that
diversify the map of the world. See Article 144.
PROBLEM VIII.
211. On the plane of the meridian, and by the stereo
graphic projection of the sphere, to draw the circles of Ion-
gitude geometrically, for a map of the world. See Plate VI
Fig. 1st.
212. With any radius C Q, describe a circle; draw EQ
and N S, at right angles to each other ; divide the quadrant
QS, into nine equal parts, to which draw the lines N 10,
N 20, &c. which will divide the radius C Q, unequally into
nine parts; or CQ is now marked as % line of semi-tangents,
which must also be set off on the radius C E.
213. Bisect CN in P, and draw PT perpendicular to

Diagrams l&xMibittint'g' tEe

Stereo graf? file <Projedwti ofthtJlciidtans.

HaZeVI
facing P<*$* £&.

----•.-IE So

Projection of an ellipse similar
to FCg. 4. Plate £$.

[/(Z/WWWW *iM-

I

aj, 2

|v|;, \
p
vM*)fi>&+rli\t*-}e NyT
ii^N, \
...-"
C
00 1 1?0 'i 7a 'i aV. if0\.ff\Jfl-\» ''Jff
*
i?
IS
: :, \ \ ¦•. '/*>
\ \ \ y™°
&
V
iW *J
¦'ofl! ."ttt^p
65
CN; then will PT bisect all the lines drawn obliquely froni
N, to the radius C Q, by Prop. IV. Orthographic Projection-
214. Secondly. See Fig. 2nd. Produce QE indefinitely
beyond E, for your line of measures, and from the points of
intersection, formed by the line PT crossing the lines drawn
obliquely from N to the radius C Q, raise the perpendiculars
10 10, 20 20,' &c. to meet CE produced. Then, these
points 10, 20, &c. in the line of measures CE produced, are
the centres, about which the respective circles, of longitude
are to be described.
215. And provided you produce CQ, as CE was pro
duced, but in the contrary direction, and perpendiculars, as
in the.former case, fall upon it, the centres for the circles of
longitude in the other semi-circle will be determined. But
as all the centres of these imagihary circles of longitude in
the.semi-circle N E S, are at the same distance from C, in the
radius C Q produced, as those in N Q S are from C in the
radius C E produced— it is obvious that they may be found,
or set off in C Q produced, by spaces equal to the distances
C 10, C20', &c. in CE produced; and not only so for this
semi-circle N E S, but also for the two other semi-circles in
the other planisphere.
• 216. Fig. 3d, Plate VI. exhibits at one view all tlie
lines in Figures 1st and 2nd, together with the meridians
drawn from their respective centres.
PROBLEM IX.
217. To draw the circles of latitude geometrically, for a
map of the world, on the plane of the meridian, and by the
stereographic projection of the sphere. See Plate VII.
Fig. 1st.
^. 218. From the centre C, with the radius C Q, describe a
circle, as in the former problem; or, if you are going to

66
complete a planisphere, you would make use of that Circle on
which you have already drawn the meridians. Divide one
quadrant, as QS, into nine -equal parts, and set off S3|° for a
tropic, and 66 j* for a polar circle; then from E draw the
oblique lines E 10, E20, &c. which will mark on CS a line
of semi-tangents, that are to be setoff on CNalso. Bisect
the segment of each of the oblique lines falling within the
quadrant Q CS, in the points c, c, &c.
219. From the points c, c, i&c. raise the perpendicular*
c 10, c20, &c. to meet the line of measures produced to
wards It ; then, on the line C R, a 10, b 20, &c. will be the
radii of the circular parallels, corresponding to the different
degrees of south latitude, the tropic of Capricorn, and the
Antarctic Circle; and about the points 10, 20, &c. on SR,
with the distance 10 a, 20 b, &c. these parallels are to be
described, as will appear obvious by a slight inspecting of
the diagram.
But the centres of the dfrcvlar parallels may also be found
agreeably to Problem A. frotn'Mngents, thus ;
220. See Fig. '2d, 2»la:te VII. from whose centre C, to the
divisions t>f (Iie'c]uadr9nt'ES,iyou drawthe-Tadii C 10, C20,
&c. ; produce the axis towards R, and from the extremities
of the radii, raise- perpendiculars, which will give in N S pro
duced, the centres of the parallels.
221. For example: From C 10, and at the point 10, is
raised the perpendicular 10 10, meeting N R in the point 10,
justR; from C20,' is raised 20 20, the point 20 in NR,
being the centre of 20* of latitude. Now, if one foot of your
compasses be placed in the point 10, in the line NR, and
the other be extended to the point 10°, in the circumference
of the plane of projection, by .sweeping across the whole
plane, the circular arc, answering to 10° of south latitude,
will be described. The remaining circles of latitude are
drawn in a similar manner, as will be perceived by glancing
at the diagram.

Plate \
tzf Irvrupap,

BMtitmci mo methods of ' a>TZftructm^ fii&(%rcle<r of latitude
I>apw<p and Tolar Circles
for ihe ttereogrdpftic jyofedzaw of -tfiey Sf>7z<

zfft '

a C s

^3«

<3

'*
I

J}raM>n, Ay -*1 Jaznieavm/.

Untpwrca

67
222. Fig. 3d, Plate VII. displays a combination of
both the foregoing methods, which we shall briefly explain.
223. By the first method, Prob. IX. for the parallel of 60°
north latitude, the segment 60 s, of the line Q 60, is bisected
in r, and from this point is raised the perpendicular rr,
meeting the produced axis in r; the point r without the
circle, is then the centre of the circular arc of 606 N. L. ;
and if about this point r, as a centre, and with the radius r
60°, there be described a circle 60*60, you would call it
the parallel of 60° N. L.
224. Again : Suppose it was required to draw the tropic of
Cancer, you bisect m 23£° in the point n, raise the perpen
dicular n n, meeting S N produced in n; and about n as a
centre, with the radius n 23§°, describe the circle 23|° m
@3§°, and it will represent the tropic of Cancer. ,
225. Moreover, it is by this method that the ecliptic in
the southern hemisphere is described-; for, let Q k be bisect
ed in d, and a perpendicular dd be raised, crossing SN pro
duced in d, you would about d as a centre, and with .the
radius d Q, describe the ecliptic E Ic Q.
,226. By the second method, i. e. by Prob. A. for the paral
lel of 40° south latitude, C 40 is a radius of the primitive cir
cle, CS is produced beyond S towards p, and from the extre
mity of the radius C 40, is raised the perpendicular 40 &0 ;
then, about 40 as a centre, in NS produced towards p, and
with the radius 40 40, you can describe the circular arc for
the parallel of 40° of south. latitude, marked on the engraving
40 south latitude.
227. Again, if you have to describe the Antarctic Circle,
(here is the radius C 66{°, b 661" for the tangent, and b in
C p, is the centre, about which, with the radius b 66±9, you
have to describe the Antarctic Circle.
228. You will observe that, in this diagram, the semi
circle ENQ, represents the northern half of the eastern
?2

68
hemisphere; and the semi-circle ESQ, is supposed to be
the southern half of the western hemisphere. It may also be
noticed, that the radii we obtained for describing the circles
of 60° N. L. the tropic of Cancer, and the Antarctic Circle,
will describe also the circles which are the converse of these,
and so on for all the others (168).
The stereographic projection of the sphere, may also be
performed on the plane of a meridian, by means of trigono
metrical tables of radii; and the two following problems ex
hibit this method.
PROBLEM X.
229. To draw the circles of longitude according to the
stereographic projection of the sphere, on the plane of the
meridian, from a trigonometrical table of radii. See
Plate VIII. Fig. 5.
230. About C as a centre, describe a circle, and through
its centre draw the diameters W E, N S, at right angles ;
the former you would call the equator, and the latter the
axis meridian. Divide the quadrant ES, into nine equal
parts ; and from N, to these divisions, draw oblique lines,
which will divide the radius C E, into nine unequal parts, in
the points a, b, c, &c. These oblique lines are given in
specimen, from E to the subdivisions, in the quadrant N W,
and the intersections made by means of them on the semi-
diameter C N, mark a line of semi-tangents, which must also
be set off on the radius C W-
231. The two poles N, S, and the semi-tangents a,b,c,
&c. on C E, mark three given points, through which each
meridian line must pass.
232. Secondly. Take the radius C E, and divide it into
ninety equal parts, of which the following table shows how
many are to be taken for the radius of each meridian.

aMqonvvuiiical wwlmetwn cfthe Glchilar and Stereo o rap i
Projedforu? of the Splure
on , t7i& Tlcuw of a tu vidian

no

¦no"/-
¦--an

263

180

¦ \ >-~

/ Kt>2?
-¦
?--''
••'n ::::-v"--X?\
W
o;
.K
- t°l - ?M*
\
O
^
t? S
to
..J —
>
>
 1-
it
ast
xid
I A GMAM 8 11". and
JBxfdtit a tecmftn'Ml eonstnielion of flat Globulzar projedww
at ' i/ie SMure on ih& pLana of a ineridian
Jhn*m. T>v j£.Jamic*tra.
Bv ,,-.-,, /., "('
69

S33. Trigonometrical Table of Radii for the Meridians, con-
, structed from a canon of Radii, as exhibited in Problem B.
page 61.

For the arc NaS the radius a a = 91-4 say 91|
 N6S -j  bb = 95^78  96

— o

._ NcS  — cc = 104  104
._ NdS  -^— dd = 117-5 — 117|
-Ne'S  ^ — ee = 140  140
- N/S — — — // = 180 — 180
— NgS  . gg = 263.  263
_ NAS  kh *= 518:3  518|J £ b

o -a
to 2
L-fi .2
O. si
S 8
O J3
.a —

234. Set one foot of your compasses in a, on the line C W,
extend the other to a, on C E, and you -will be able to draw
the meridian NaS; and so on for any of the others.

PROBLEM XI.
235. To draw the circles of latitude stereographically on
the plane of the meridian, from a trigonometrical table of
radii. See Fig. 6th, Plate VIII,
236. You construct the plane of projection, as in the last
problem, and then divide the quadrants NE, N W, each
into nine equal parts, to each of which, as well as to 231" and
66\°, there are drawn lines front the points W and E, and
the intersections made by them, as seen on the semi-diameter
C S, mark a line of semi-tangents, which are denoted on C N
by the letters r, s, A, t, &c. And here we have three points,
through which to describe the parallels of latitude; namely,
a point r, s, &c. on C N, and the corresponding divisions on
the arcs W N, and E N. The centres of these parallels, will
be in the line S N produced.

237. If the radius CN, bo divided into ninety equal parts
for a scale, then the following table shows the length in those
parts of ihii scale, which you take to describe each arc.
Trigonometricql Table bf Radii for Latitudes, ConStructedfrofif
a canon of Radii, as Qxemplified in Probleni C. page 63,

Forthearc '80r80 thpradiusis 'r r. = 15.87 say 16.
Ji  :  70s 70 —  . ss == 32-75 — 32|
 : — 66§A66i—  A. Arctic = 39-19 — 39
.  '¦  60*60  tt =51-96 — 52
 '¦  SOoSo ,— :  vv = 75:S2 — 751
-^-  s- 4bw40  . tow =• 107-3' — 107|
 ¦ 30a; 30  x x = 1559 — 156
¦  23|T?3|  T.Tropic= '207 — 207
 20 #'20  . yy = 247-3 — 247£
,  JO a 10 — r— -. — . xz = 510-4 — 510|

a ts
O.I-.

o *¦
J3

238. If, then, you set off these distances on SN produced,
and extend your compasses from r to r, you will be able to
describe the circular parallel 80° r 80p ; and so on for any
of the rest,

PROBLEM XII.
239. To make a stereographic projection of the sphere, on
the plane of the equator. See Plate X,
240. Describe the circle NWSE, with any radius, and
draw the diameters W E? N S, at right angles to one another,
Divide the arcsi of each of the four quadrants, into nine
equal parts ; subdivide each of those parts into 10®, so
shall the whole equator be divided into 360°.
241. Ttien, from P, the pole, or centre of the ^circle, to
each of the primary divisions in the equator, draw radii,

t

^

:^Ǥr

S Br

71
which will represent the meridians projected into straight
lines, as P 10, P 20, P 30, &c.
242. For the parallels of latitude, project a line of semi-
tangents, thus* from W, to the divisions 10, 20, &c. in the
quadrant EN, (or 90,) draw straight lines, which will di
vide the radius PN into a line of semi-tangents, through.
the intersection of which, describe, about P as a centre, the
circles of latitude 10, ^0, &c. number the degrees of lon
gitude and latitude, and the projection will be completed.

The following rule may be employed for determining the
size of a globe, upon which the countries shall be of the same
dimensions as you see them in any map of the world.
243. When in any map the degrees of latitude on the pri
mitive circle are Just one inch in length, the globe, of which
that map represents the surface, will be ten feet in diameter.
The reason is obvious, by considering that the circumference
of that globe will be divided into 360 inches, each degree of
the circumscribing circle being one inch, in !length~; and as
the diameter of every sphere is one«third. part of its circum
ference nearly, JSfi- « 120 inches,, which, divided by 12,
give just 10 feet; and, therefore* a map, on which the de
grees of latitude are f an inch in length, will he the part of
the surface of a globe half so much in diameter, or 5 feet.
If the degrees in a map are T«*h of an In4;h, the diameter of
the globe will be 12 inches. On the other hand, if the de
grees of latitude are two inches on the map, then such a map
is part of the surface of a globe 20 feet in diameter; and so
in proportion for degrees of any other length*.

* "Young Gent, and Lady's Philosophy," Vol. 2d. Dialogue XI V.
Page 134, Edition, 1763.

n

SECTION V-
THE PRINCIPLES OF MERCATOn's PROJECTION, WITH A,
COPIOUS ILLUSTRATION OF THIS PROJECTION, AND PRO
BLEMS EXHIBITING THIS MAP CONSTRUCTED  FIRST,
FROM A TABLE OF MERIDIONAL PARTS; AND, SECONDLY,?
GEOMETRICALLY; AND A, METHOD OF ENLARGING OR
JtEDCCJNG MAPS, PROPOSITION I.
244. If P be thepole of the earth, E Q the equator, P E,
PR, two meridians, mn a small circle parallel to ER ; then
the length of. a degree of latitude is to the length of a de
gree of longitude at m, as radius is to the cosine of the
latitude at m, supposing the earth to be a sphere. See
Fig. 2, Plate XII.
245. For, let PC be the radius of the earth; draw
mr, nr, perpendicular to it, and join EC, RC. /Then mr,
n r, being parallel to EC, R C, respectively, the angle mm,
is equal to the angle, ECR; hence, by similar sectors, E R i
mn: : EC : mr, the cosine of m E. But when the angle
is given, the length of arc of a degree, is in proportion to the
radius; also, the length of a degree of the great, circle
ER, is a degree of latitude; and the length of a degree of
the circle mn, is a degree of longitude at m. Hence, a de
gree of latitude j a degree of longitude : : radius : the cosine
of latitude*. .

* See page 227, in the " Principles of Fluxions," hy the Rev. S. Vince,
AM. F.R.S. 3d Ed. 1805. See also Emerson'} " Math.' Prjn. of. Geo."
referred to page 74.

73
246. In Mercator's- projection, the sphere is projected
upon a plane; and th^-meridians E P, RP, are projected into
straight lines, and parallel to each other; consequently, P
in the projection, must be at an infinite distance from the
equatorEQ. In this case, the arc mn being the same for
all latitudes, the length of a degree of longitude is every
where the same. To preserve, therefore, the proper pro
portion between the- degrees of latitude and longitude, the
degreqs of latitude must increase as you go from the equator,
go that they may always be to the degrees of longitude, in the
proportion of radius to the cosine of latitude, \"

PROPOSITIQN II.
¦ . In this. projection, it is r£quired to find the length of an
arc of the meridian, correspdnding to any given latitude.
See Fig. 3, Plate XII."
247. Let P be the pole, E the equator, P C Q the axis of
the earth, C the centre, m any place on the surface ; draw
mr perpendicular to PQ; and join ut C, w Q. Put Cm
i=i= r,Em= x, C r (the sine of E m the latitude of m) = y,
and the length of Em on the projection = %, called the
meridional parts. Then, (by Prop. I.) V r* — yz (cos of lat.)

r x

• ; but * x = - r '^— ; hence,

V r*— yx ' V rl—yz
z = — ,-J-a. =_i- v — r-M—- therefore, z = 77 ' X !•¦ !•
r*— r/* 2 rz—yx ' 2 .

* By Art. 46, in " The Principles of Fluxions," To find Fluents by
Circular Arcs, Prop. XIX.

n ,¦*¦ ¦<¦•-
- — - -f cor. *=>rxIi.L V ^ ^ + cor., by the nature
r—y r—y J
-¦ -ii ) ¦ -i
of logarithms. But, by Plane Trigonometry, V rz — y*
fXf + y
(bit) : r + y (r Q) : : r (rad) : /— r-. _, ~€ *»
r V  the tangent of the angle rwiQ = cotangent
of r Qm — cot. of f r C m = cot. of half the complement of
, , „ /r + y cot. | comp. lat.
latitude. Hence, V — —£¦ == - — — —  ' ¦•'r" i con-
., , , cot. | comp. lat.
sequently, s = r x h.I.  + cor.; but
when s = O, cot. f comp. lat. = r ; therefore, the last equa-
r
lion becomes O == r X h. 1. — -f cor. = rxi.l. 1 +
r
cor. = O + cor. therefore the correction is = 0 ; con*
, , cot. \ comp. lat.
sequently, * = r X h. 1. — -¦¦ r  =rX h.I. cot.
1 comp. lat. — r X h. I. r, the length qf the meridian Em
on the projection. If, therefore, we take the latitude =
1°, 2°, &c.  90% we can construct a table, showing the
length of the meridian on the projection, for every degree
of latitude. In like manner, it may be constructed for every
minute+. Siifeh a table, is called a table of meridional parts.

* Art. 45. Ex.6. Prop. XVIII. Tojindtliejluentofajluxionrtt;hick
isthefit&cimtdfwiy quaatitp'ty) timidbd by that quantity (yy,-6r in a given-
ratio to it.
f See the article " Navigation," in Emerson's Math. Prin. of Geo
graphy, octavo, London, 1770,

w

248\ Table of Meridional Paris for every Degree, from 1*
to 84°, north of south of the Equator.

i  ,
Deg.

Mer. Pfe.

Deg. 56

Mer.Pts.

Deg. 28

Mer. Pts.

c
84°

10137

4074

173 1

83

9606

55

3968

§7

1684

82

9145

54

3865

26

1616

81

8739

53

3764

25

1550

SO

8375

52

3665

24

1484

~ 79

8046

51

3569

231

1451

78

7745

50

3474

23

1419

77

7467

49

3382

22

1354

79

7210

48

3292

21

1289

75

6970

47

3203

20

1225

7i

6746

46

3116

19

1161

73

6534:

45

3030

18

1093

72

6335

44

2946

17

1035

71

6146

43

2863

16

673

70

5966

42

278*

15

910

69

5795

41

2702

14

848

68

5631

40

2623

13

787

67

5474

39

2545

12

725

66\

5398

38

2468

11

664

66

5324

37

2393 ]

10

603

65

5179

36

2318

9

502

64

5039

35

2244

8

.482

63

4905

34

2171

7

421

62

4775

33

2100

6

361

61

4649

32

2028

5

360 "

60

4527

31

1958

4

240

£?

4409

30

1888

3

180

58

4294

29

1819 ,

2

120

57

4183

-B

egin at 1

M

60

and go

up the table t

ill you cc

>mc ((

) 84°, or

the po
' ' ' V.' '

int C.

76
249. If we take the earth of its true figure, that of a sphe*
roid, we may compute the meridional parts upon the same
principles; but we shall not here give the investigation, as
we only want to explain the nature of this projection, so far
as it may be necessary, to show how the charts are con
structed. If we assume Sin Isaac Newton's ratio of the
diameter of the earth for 50° latitude, the difference of the
meridional parts on a sphere and spheroid, will not be above
the 60th part of the whole. It is manifest that this projec
tion cannot give the true position of the parts of the earth ;
for the figure of the part ER, mn, (Fig. 2nd, Plate XII.)
on the projection, would be a parallelogram. It is, how
ever, very convenient for navigation, because the rhumbs
are all projected into straight lines ; for, the meridians being
all straight and parallel lines, the line which cuts them all
al the same angle, must be a straight line, which is the proi
perty of the rhumb. See Section VI. article Rhumb Line.
250. Mercator's projection is chiefly confined to charts for
the purposes of navigation. In this projection the meridians,
parallels, and rhumbs, are all straight lines; but, instead of
the degrees of longitude being every where equal to those of
latitude, as is the case in plain charts, the degrees of latitude
are increased as we approach towards either pole, being
made to those of longitude in the proportion of radius to the
sine of the distance from the pole, or cosine of latitude; or,
what is the same thing, in the secant of the latitude to radius.
Hence, all the parallel circles, are represented by equal and
parallel straight lines ; and all the meridians, as has already
been noticed, are parallel lines also ; but these increase inde*
finitely towards the poles *.

* Encyclopedia Britannica, article, Mercator's Projection, under
the word Geography. See also Martin's Mathematical Institutions,
article, "Navigation," Vol. 2d. Edition, 1764. London,

77
251. From this proportional increase of the degrees of the
meridian, it is evident, that the length of an arc of the meri
dian, beginning at the equator, is proportional to the sum
of all the secants of the latitude ; or, that the increased meri
dian bears the same proportion to its true arc, as the sum
of all the secants of the latitude, to as many times the radius.
The increased meridian is also analogous to a scale of the
logarithmic tangents, though this is not at first very evident*.
252. The meridian line on Mercator's chart, is a scale of
logarithmic tangents of half co-latitudes. The differences of
longitude on any rhumb, are the logarithms of the same
tangents, but of a different species; those species being to
each other, as the tangents of the angles made with the meri
dian. Hence, any scale of logarithmic tangents, is a table
of the differences of longitude to several latitudes upon some
one determinate rhumb ; and, therefore, as the tangent of the
angle of such a rhumb : tangent of any other rhumb : : the
difference of the logarithms of any two tangents : difference
of longitude on\ the proposed rhumb, intercepted between
the two latitudes, of whose half complements the logarithmic
tangents were taken t.
253. Mercator's projection is exhibited by Fig. 1st,
Plate XI. where a fourth of the chart of the world is
drawn, from the equator to 84° north, and 90° east and west
of London.
PROBLEM I.
Of Drawing the Meridians for this Map.
To Draw the Meridians and Parallels for Mercator's Chart
of the World.
254:. First. Draw the line A B, representing the equator,
bisect this line in o, and erect the perpcndicnlar o o for the
* Note I. f Note K.

78
meridian of London. Assume a distance fojf 10°, whish get
off from p towards A nine times, and from o towards B nine
times; numbering both ways 10°, 20?, 3Q°, fyc Then,
from the points fy. and B, draw indefinitely AC, BDj parallel
to oo„ the meridian of London; and through tlie points J.0,
20, 30, &c on either side of o, draw lines parallel tp A C or
B D, which will be the meridians. On the two lines A C, B D,
are to be set off the parallels of latitude.
255. In order to draw these parallels of latitude, it will be
necessary to construct a diagonal scale of equal parts, 60, of
which shall be equal to 10° on the equator; consequently,
one part will be equal to one-sixth of a degree, or to ten
geographical miles, or to ten meridional parts. The con
struction of this diagonal scale depends immediately upon
a geometrical problem, which teaches to cut off the successive
parts of a given straight line*. The diagonal scale is of
the most extensive use, and constituted the first improvement
On astronomical instruments.
Of the Construction of a Diagonal Scale, by which to Draw
the Parallels of Latitude for this Map.
256. Draw A C (Fig. 2d, Plate XL) equal to the dis
tance assumed for fen degrees on the equator in Fig- 1st,
Plate XI. and considering AB (=AC) as a division,
containing 10° of the extreme portion of the equator,
set off sixty times from A to B, and divide A C into ten
equal parts, each of which is again subdivided into sixty
secondary parts, by means of ^the diagonals which decline
from A B by intervals equal to the primary divisions, .and
which are cut transversely into ten equal segments, by equi
distant parallels. In article 256, we have constructed a

* LeslieVGeom. B. VI. Prop. 5, -Second -Edition.

79
diagonal scale, from Gunter's line of equal parts, in order
that you may construct Mercator's chart by means of that
diagonal scale (Fig. 3d), and the meridian line on Gunter's
scale. 257. The meridian line on Gunter's, scale, is nothing but
tlie table of meridional parts in Mercator's projection trans
ferred on a line, which may be done in the following man
ner, by means of the line of equal parts set under it, and a
table of meridional parts*. See Plate XL Fig. 3d.
258. Thus, take any one of the large divisions from the
line of equal. parts on Gunter's scale, known by the letters
EP+, signifying equal parts, and on the scale set under the
line of meridional parts, and let this division from the scale
of equal pa»ts, be the estreme portion of a .horizontal line,
which is to be divided into sixty equal parts. Let A B,
Fig- 3d, Plate XL be -the length of one of those divisions^
and divide it into six equal parts; then, at the points AB,
perpendiculars A.C, BD, are raised equal in length to AB,
and the line D C completes the parallelogram ABCD.
Each:of the sides AC, BD, is divided into ten equal parts,
and these are virtually subdivided into secondary parts by
the diagonals A F, 10 20, &c.
259. Yon will now have a diagonal scale, by which any
part of the aforesaid division, under 60, may be easily taken ;
and it is obvious, that this subdivision is effected by means
of the diagonal lines, which decline from the. perpendicular,
by intervals equal to the primary divisions, and which are
cut transversely into equal segments, by equidistant parallels.
Each,j)rimary part on the diagonal scale, contains ten meri
dional parts, or 10 geographical miles (253.)

* See Bion's Const, and Use of Math. Tnstr. by Stone. 2nd Edition.
Folio. Page 41. Fig. 9. Plate V. London, 1758.
f It will be perceived, tljat in this instance I used one of the old scales.

80

Of Drawing the Parallels of Latitude in this Map.
260. Now, supposing that you are to draw every degree*
of latitude north and south from the equator to 84", for
Mercator's chart cf the 'world, as these degrees of latitude
must be set off by unequal divisions of the meridian line,
you look in the foregoing table of meridional parts, for 1#,
and over against it you will find 60, and rejecting the last
figure, or unit's place, which' in this case is o,> take 6 equal
parts from the aforesaid diagonal scale, and lay it off on the
meridian line, which will give the division for one degree of
latitude. Again, to find the division for 2", seek in the table of
meridional parts, for the number opposite to 2°, and you
will find 120; striking off the units's place, (which is always
to be done,) take 12 from your scale, and lay^t off from the
beginning of the meridian line, and the division for 2° will
be had.
Moreover, to find the division for 11°, you will find
answering to it 664, and rejecting the last figure, the remain-
der'66 laid off from the beginning of the meridian line, will
give the division for 11°. But, because 66 cannot be taken
from the diagonal scale, you must take only 6 from it, and
for the 60, take its whole length, or else lay off the six from
the end of the first division of the line of equal parts, and the
division for 11° will be found.
In this manner may the meridian line be divided info
degrees, and every SO' as it is upon the scale. There are
several other methods of dividing this meridian line;. but let
the examples we have given suffice, remembering, that the
great use of the meridian line on Gunter's scale, is to project
and use M creator's chart.
261. The meridians and parallels of latitude having now
been drawn to every ten degrees of the map, Fig. lsts

C 80

¦W

Merc-ator's J^ojection
60 SO 4.0 30 20 20 c

10

Fhiladdfihia,

go SO JO
JarmJtcrn cfeL'.

I

JhtaOrVo

ZandAn

sPan?

^€Hralta.

ZC L J? d0 J° $° 7° So- T>

on&vm

fettr. ^f&wpv

Consta* vUrwp't &

So

1° 65 60
JO
as ao

\3S 30 2J

20TS 10
-J

JO 40 30 20
/tquaUrr

jtteridtanoFZtm&ort,

20 SO /W SO 6b /O So go
Mff2 Smart sculp,

hale XI

81
Plate XI. I have only to observe, that if you project a
map of this kind on a scale of one inch to each 10° on the
equator, till the objects that diversify the map are inserted,
pencil lines may be used, being drawn through every indi
vidual degree ; and a mariner's compass may be placed in
some convenient place that is otherwise blank, together with
a scale of geographical miles.
262. When the map is thus prepared, any place may
easily be inserted in it, if the latitude and longitude are given.
For example: let it be required to lay down the following
places in the map, according to the latitude and longitude
affixed to each.

North

Lat.

Longitude.

51°

31'

0°

EDINBURGH 
55°
27'
3°
7'
W
53°
21'
6°
1'
—
53°
59'
1°
1'
—
Paris .... 
48"40"
50'25'
2° 3°
25' 20'
E.
W-
48°
12'
16e
22'
E.
Constantinople
41°
1'
28°
58'
—
Philadelphia .
39"
56'
75°
9'
W.
Gibraltar 
36°
5'
5°
17'
— ¦
Jerusalem 
31°
11'
30°
21'
—
Petersburgh . .
59°
56' .
30°
24'
—
9.63. The latitude of London is 51° 31' north; but as the
scale is small, it is difficult to set off the odd minutes exactiy,
and a few more or less than the actual number, is an error of
no magnitude to affect the work. In this example we shall
call London 51|" N. Lat. which being set off on AC and
B D, at a and a, lay your ruler from a to a, and the point on
the meridian of London, is its situation.
G
82
264. Again. The latitude of Edinburgh is 55" 57', say
56°, which set off at n, n, on the lines A C and B D. The
longitude is 3° 12' west of the prime meridian, or London,
which set off on the lines CD and AB, and draw a line,
which will determine the situation of Edinburgh; for, where
two straight lines drawn from n to n, and from r to r, inter
sect each other, there the place is situated ; and in the same
manner may the situation of any other place, or the mouth
of any river, be determined. But we must now refer the
student to the objects that diversify the face of a country,
and the waters of the deep, for the materials with which to
fill up his map. See Section II. p. p. 23 — 30.
PROBLEM II.
To project Mercator's Chart geometrically. See Fig. 1st,
Plate XV.
265. 1st. Draw the line AB, of any convenient length,
which divide into nine equal parts ; then, at the extremities
A and B, raise the perpendiculars AC, B D, and through the
points of division in AB, draw the other eight meridians
parallel to A C or B D. These will now be the meridians
for one eighth part of a map of this nature.
266. 2d. Set one foot of your compasses in A, extend the
other to B, and sweep the quadrant B F. Divide B F into
nine equal parts, and from A to each of these, draw the radii
A 10, A 20, &c. You will observe, that these radii cut off
unequal spaces on the line G H, and that the lines A 1, A 2,
A 3, &c. are the distances of the parallels from one another.
267. 3d. Take the line Al in your compasses, and from
the points A and B, set off the parallel of 10 degrees; then,
take the distance A 2, and placing one foot of your compasses
in the point I, and set off IK, whieh will be the parallel of
20 degrees. Again, as A3 is equal to KL, A 4 =LM,

83
A5 = MN, A6 = NO, A7=OP, A8 = PCandPD,
you may with the utmost ease determine and draw the eight
parallels for this diagram.
268. This construction is simple and beautiful, but it is
only an approximation to the truth. It is, however, so very
easy of execution, for drawing general maps, or finding the
points of each tenth parallel, that where there is not the
greatest accuracy required, I believe the young geographer
will adopt it in preference to that in Problem I.
Other Methods.
269. Mercator's Chart may also be projected by a line of
secants on the sector; or by a line of lines on the sector; by
Gunter's scale, from the line of meridional parts ; all which,
with Problem II. geometrically, may be considered as four
methods of constructing a nautical map instrumentally ;• and
if you add to these Problem I. which is an arithmetical pro
jection from the table of meridional parts, there will be five
methods by which this projection may be effected.
Of enlarging or reducing 3Iaps.
270. A diagonal scale will be found very convenient for
enlarging or reducing maps in any given proportion. For,
example: Suppose it were required to enlarge a map one-
half, one-third, or one-fourth ; say that we enlarge our map
in the ratio of four to one; draw the line A B, (Fig. 4th.
Plate XI.) and having taken 10° on the given map within
your compasses, set off on the line A B, five equal parts,
each = 10° in question. At the first division C, erect the
perpendicular C E ; A C is = |th of C B. Next, take any
convenient distance in your compasses, and set off any num
ber of equal parts on the perpendicular; for example, 60;
g2 ,

84
and through each point of subdivision in CE, draw a line
parallel to A B, and draw the two diagonal lines, one from
A, and the other from B, intersecting each other at the same
point E.
271. Measure any distance in the original map, and apply
it to the smallest scale, and find which line among the paral
lels it exactly coincides with ; keep the point of your com
passes on the perpendicular; turn the other point over, en
larging them at the same time, till the other foot falls into
the diagonal BE; and this corresponding parallel on the
larger scale, will be four times the original quantity; or you
will have enlarged your map in the ratio of four to one; for
all the other distances are to be taken in a similar manner.
272. It is evident, that to reduce a map one-half, one-
third, or one-fourth, the operation is precisely the opposite
to this ; for, in diminishing your map, the compasses would
be applied to the larger scale first, and turned over on the
perpendicular, and then closed to coincide with the extreme
points of the corresponding parallel on the smaller scale;
and so on for every distance taken in diminishing the map.
273. But after all that can be said about scales for reduc
ing or enlarging maps, there is nothing so effectual and so
sure for this purpose, as the proportional compasses. To
those who know the method of using this admirable instru
ment, nothing need be said; but, as there may be some to
whom even the name of such an instrument is altogether
new, the following note may be of very considerable ser
vice*.

* See Note L.

85

SECTION VI.

OF THE ORIGIN AND PROPERTIES OF THE RHUMB
LINE, &C.
274. A HE rhumb line, loxodromia in navigation, is a
line prolonged from any point of the compass in a nautical
chart, except the four cardinal points ; or it is the line which
a ship, keeping in the same collateral point or rhumb, de
scribes throughout her whole course.
275. The chief property of the rhumb line is, that it cuts
all the meridians at the same angle ; and this angle is called
the angle of the rhumb, or loxodromic angle. But the angle
which the rhumb line makes with any parallel to the equator,
is called the complement of the rhumb.
276. An idea of the origin and properties of the rhumb
line, the great foundation of navigation, may be conceived
thus : — A vessel beginning its course, the wind by which it
is driven, makes a certain angle with the meridian of the
place ; and as we shall suppose that the vessel runs exactly
in the direction of the wind, it makes the same angle with the
meridian which the wind makes. Supposing, then, the wind
to continue the same, as each point or instant of the progress
may be esteemed the beginning, the vessel always makes the
same angle with the meridian of the place where it is each
moment, or in each point of its course which the wind
makes. 277. Now, a wind, for example, that is north-east, and
which consequently makes an angle of 45° with the meridian,
is equally N. E. wherever it blows, and makes the same
angle of 45° with all the meridians it meets ; and, therefore,
a vessel driven by the same wind, always makes the same

86
angle with all the meridians which it meets on the surface
of the earth.
278. If the vessel sail north or south, it describes the great
circle of a meridian. If it runs east or west, it will cut all
the meridians at right angles, and describe either the circle of
the equator, or else a circle parallel to it.
279. But if the vessel sails between the two, it does not
then describe a circle, since a circle drawn obliquely to a
meridian, would cut all the meridians at unequal angles,
which the vessel cannot do. It describes, therefore, another
curve, the essential property of which is, that it cuts all the
meridians in the same angle ; and it is called the loxodromy,
or loxodromic* curve, or rhumb line.
280. This curve on the globe, is a kind of spiral, tending
continually nearer and nearer to the pole, and making an
infinite number of circumvolutions about it, but without ever
arriving exactly at it But the spiral rhumbs on the globe,
become proportional spirals in the stereographic projection
on the plane of the equator.
281. The length of a part of this rhumb line or spiral,
then, is the distance run by the ship while it keeps in the
same course. But as such a spiral would prove very per
plexing in the calculation, it was necessary to have the
ship's way in a right line, which, however, must have the
essential properties of the curve line; viz. to cut all the
meridians at right angles.
282. The arc of the rhumb line is not the shortest dis
tance between any two places through which it passes ; for
the shortest distance on the surface of the globe, is an arc of
the great circle passing through those places ; so that it
would be a shorter course to sail on the arc of this great

* The word is from the Greek, and signifies ah oblique dourse.
Dr. Johnson's Dictionary gives the best definition to be met with.

87
circle i but then the ship cannot be kept in the great circle,
because the angle it makes with the meridians, is continually
varying more or less *.,
283. Let EQ (Fig. 4th, Plate XII.) represent the equa
tor, let P be the pole, and let P E, PR, be two meridians ;
also let the spiral, or curve line A smv, represent a rhumb ;
then will the angle P s m, be equal to the angle P m v. Draw
now the small circle n m, parallel to E Q ; and let P E, P R,
be conceived to be indefinitely near one another ; then you
may look upon the triangle s m n, to be a plain triangle :
hence sm : sn : : Rad. : cosine nsm;
but the angle nsm, is constant for the same rhumb, and there
fore sm is to sn, in a constant ratio; in other words, as the
rhumb A smv, approaches towards the pole,, the increment
of the rhumb is to the corresponding increment of its latitude
in a constant ratio ; therefore, by composition, the whole
increase of the rhumb, is to the whole corresponding increase
of latitude in the same constant ratio -f .
284. When, therefore, a ship runs upon the same rhumb,
and changes her latitude five degrees, for example, if she
continues on the same rhumb, and describe the same space,.
it is evident that she will again have altered her latitude five
degrees. 285. It may be observed generally, that equal parts on the
same rhumb are contained between equidistant parallels of
latitude. 28g. The rhumbs, or points of the compass, have each
a particular denomination, expressed by means of the initials
of the four first points, North,. East, South, and West, viz.

* Dr. Hutton's Mathera. Diet. vol. 2d, p. 373. Edition, 1795.
f Vince's Astronomy, art. Rhumb Line; and No. 93 of La Croix'fi
Crit. Geog,

88

North.

East.

South.

West.

N. by E.

E.byS.

S. by W.

W. by N.

N. N. E.

E. S. E.

S. S. w.

W.N. W.

N. E. by N.

S. E. by E.

S. W.byS.

N. W.byW

N. E.

S.E.

s. w.

N.W.

N. E. by E.

S. E. by S.

S. W.byW.

N.W.byN.

E. N.E.

S. S. E.

w.s. w-/

N.N. W.

E. byN.

S. by E.

W.byS.

N.by W.

Each of these points contains 11° 15', which are again
divided into \ points, containing 2° 48' 45".
It was the Greeks who at first gave only four names to the
winds, corresponding to the four points, called afterwards the
Cardinal Points. They afterwards added four others, corre
sponding to the four points in which the sun rose and set at
the winter and summer solstices. Seneca, in his Questiones
Naturales, has given their names ; and the inquisitive student
will find in No. 91 of La Croix's " Introduction to Critical
and Mathematical Geography," a beautifully classical ac
count of Eurus, Zephyrus, Boreas, and Notus, among the
ancients, and the assemblage of the thirty-two divisions of the
modern well-known mariner's card.

The method of determining the Rhumb upon which a ship
sails, in going from one place of the world to another, ex
hibited on a Mercator's partial Map.
287. The very full manner which I have entered on this
projection, renders it unnecessary to add much more ; but I
shall shew you now, that the most convenient way for a ship
in traversing the ocean, and going from one place to another,
is always to sail upon one point of the compass, or, in the
language of a sailor, upon the same rhumb. The rhumb
line has been illustrated at the beginning of this section, and
we have seen in the construction of Mercator's chart, that
it consists wholly of straight lines, and no others are necessary

89
for its use ; we can, therefore, determine immediately upon
what rhumb you are to sail.
288. For, suppose that A B, (Fig. 10th, Plate XII.) re
presents 4° of longitude, lying between the seventh and
eleventh, that AC, perpendicular to it, contains 4° of lati
tude, extending from 30° to 34° : the former - degrees are
equal one to another, but the latter are increasing, as was
shown by article 246 ; they are, therefore, to be laid down
by a table of meridional parts (245, 248, and 249).
289. Let us now conceive by this partial hydrpgraphical
map, that a ship is sailing from a, in longitude 7° and lati
tude 31°, to b, in longitude 10° and latitude 33°, and you
were required to find the rhumb she must sail upon; you
have only to join a b, and that is the rhumb.
290. The method of determining what this rhumb is, is
equally simple. There is always in these maps, one or more
points, from which are drawn the thirty-two points of the com
pass, as you find in this diagram, or on Plate IV. and you
may easily discover to which of those lines, or nearest to
which a b is parallel, and thus you obtain the rhumb you
are to sail upon *.
291. But you will observe, that rhumbs are denominated
from the points of the compass in a different, manner from
the winds : thus, at sea, the north-east wind is that which
blows from the north-east point of the horizon towards the
ship in which we are ; but we are said to sail upon the north
east rhumb, when we go towards the north-east.
292. The readiest and most useful instrument for this pur
pose, is a parallel ruler ; for, by laying one edge to coincide
with a b, and bringing the other edge over the point from
which the lines of the compass diverge like so.mahy radii,
you in an instant find what you seekt.

* See Note M.
f See Emerson's Math". Princ. of Spherical and Spheroidal Sailing,
Section III.

90

SECTION VII.

MERIDIONAL, EQUATORIAL, AND HORIZONTAL CONSTRUC
TIONS OF MAPS*.
293. W E have already shown that a map is the repre
sentation of the whole or a part of the surface of the earth;
and to be perfect, it ought to exhibit, first, the true latitude
and longitude of every place ; secondly, all the countries on
such a map ought to be of their proper figures' and magni
tudes; and, thirdly, the relative situations of all the countries
ought to be truly laid down. But it is impossible that all
these circumstances can take place on a plane surface of any
construction. No map upon a plane, can then be a true
representation of the countries upon the earth's surface; but
representations may be obtained sufficiently exact for all the
purposes of geography.
PROBLEM I.
To project the planispheres orthographically for a map
of the world, in the position of right a sphere. See Fig. 5th,
Plate XII.
294. Suppose that this figure represents to you the ortho
graphic projection of the earth, upon the plane of a meri
dian, in the position of a right sphere ; then would P repre
sent 'one pole, and p the other; and the equator would be the
lineEQ.  '
* NoteN.

91
295. The meridians, you will perceive, are all projected
into ellipses (128), whose semi-minor axes Oa, Ob, Oc,Od,
O e, are the cosines of the meridians from P E p Q. And, as
the cosines of the meridians near the plane of projection
differ but slowly, those meridians will there be crowded
together, and the figures of the countries will be very much
contracted in longitude.
296. The eye at an infinite distance from the surface of
the sphere, as is the case in this projection, would view the
parallels of latitude perpendicular to the plane of projection ;
they would, therefore, be projected into straight lines in their
proper proportion (126).
297. The fundamental properties of this construction may
be demonstrated geometrically as follows. See Fig. 2nd,
Plate XX.
298. First. Describe a circle of any radius O W, draw
the diameters W E and NS, at right angles to each other,
and divide the quadrant WN, into nine equal parts; then,
as the eye in the direction of S, at an infinite distance from
the surface of the sphere, would view the rays of light pro
ceeding from the several divisions 10, 20, 30, &c. in WN,
as right lines, parallel to NO, and consequently parallel
among themselves, the projection of the ray corresponding
to the 80th degree, will be distant from O, by the space O w,
equal to n 80, the sine of 10° from N. But, on the other
part of the hemisphere, the projection of 10°, by the ray
10^>, is no more than the line W p, which is the versed sine
of 10°, and is very small in comparison of Ow in the middle
of the planisphere* If you, in executing this projection,
draw through the points 10, '20, 30, &c* lines parallel to
N S, they will divide the radius O W, in the proper propor
tion. The space Op, or O q, or O w, comprised between
the diameter and the visual ray 10^, or 20 q, or 80 w, is
the semi-conjugate axis of the ellipse, which this circle has
for its projection j and the great or transverse axis is the

92
diameter of the sphere, or of the first meridian, which remains
circular. If ellipses (303) be described through the points
w, c, u, &c. in the radius O W, the meridians shall be
described in the semi-circle NWS, for an orthographic
projection of the sphere on the plane of a meridian.
299. Secondly. To draw the parallels to the equator, whose
planes are perpendicular to the first meridian, draw through
the points 10, 20, 30, &c. in the quadrant WN, straight
lines parallel to W E, and the parallels of latitude shall be
described as they appear by the lines / 10, g20, h 30,
i'40, &c. in the quadrant WNO.
300. Thirdly. If the meridians were required to be drawn
through every fifteenth degree, divide the quadrant ES,
into six equal parts ; and, as the eye at an infinite distance
from the surface of the sphere, in the direction of N, would
view all the rays proceeding from the points 15, 30, 45, 60,
75, in ES, as right lines parallel to S N ; if you draw the
parallel lines 15 V, 301V, 45 III, 60 II, 75 1, the projection
of 15" in the middle of the hemisphere by the ray of 75 I,
will be equal to Ol, equal to 75 e, the sine of 15°. But,
on the other part of the hemisphere, towards E, the pro
jection of 15°, by the line 15 V, is no more than V E, which
is the versed sine of 15°, and is very small in comparison of
O I in the middle. The elliptical meridians drawn through
I, II, &c. constitute the hour-lines on the Analemma, be
cause 15° is equal to one hour of time, by article 39.
301. Fourthly. WeTe the parallels required to be drawn
for every fifteenth degree of latitude, then through the points
15, 30, 45, &c. draw lines parallel to the diameter WE,
and what was required will Be done. These parallels arc
seen in the lines a 15, b 30, c 45, &c.
302. The following problem for drawing an ellipse, will,
I presume, in the construction of orthographic maps and the
Analemma, be of considerable service to the young geor
grapher.

93

PROBLEM D.
Let it be required to find any number of points in the
curve of the ellipse denoted by c$ and through those points^
lo describe that ellipse. Fig. 5th, Plate XIX.
303. Let Pp be the transverse axis, and c O the semi-con
jugate axis ; let it be required to find any number of points
through which the ellipsis denoted by c shall pass. On
the transverse axis Pp, describe a semi-circle PEjj ; and
with the radius O c, describe the semi-circle GcF; take
any number of points nm, &c. in the circumference PEp,
and draw nO, mO, &c. cutting the semi-circle GcF, at a
and r; from the points n, m, &c. draw lines nb, ms, &c.
perpendicular to Pp; also, from the points a, r, &c. draw
lines ab, rs, &c. parallel to Pp; then the points b, s, &c.
are in the elliptic curve, and thro tfgh these points may an
ellipse be described. When the points for half the curve
are found, the corresponding points for the other half will
be readily obtained, by producing the perpendiculars to the
other side of Op, and making the ordinates on the one side
equal to those on the other*. (See the Appendix, art. Ellipse.)
PROBLEM II.
To project the planispheres orthographically on the plane
of the equator, in the position of a parallel sphere, for a map
of the world. See Fig. 6th, Plate XII.
304. You can conceive that, in this figure, the projection
is made upon the plane of the equator E QUA, and as your
eye is supposed to be situated at an unlimited distance be-

* See Mr/ P. Nicholson's Architectural Dictionary, article, Ellipse,
p. p. 424—436.

94
yond the pole, the meridians would form so many radii,
whose common termination inward would be the pole, and
the parallels of latitude would be circles concentric with
EQUA.
305. To form an idea of this, it is sufficient to inspect
Fig. 2d, Plate XX. (analogous to Fig. 6th, Plate XII.)
the visual rays 10 p, 20 q, &c. drawn by the different points
of the circle E N WS, considered as the equator, will deter
mine on its radius O W, (equivalent to P A, Fig. 6th,
Plate XIL) the graduation conformably to the laws of the
projection. The distances Op, Oq, Or, Os, Ot, Ou, Ot,
O w., are, therefore, the radii of the concentric parallels of an
orthographic map on the plane of the equator. Thus do
the lesser axes of the elliptical meridians of the meridional
projection, become the diameters of the concentric parallels
of Abe equatorial projection,
306. But you will perceive that as these concentric circles
are diminishing slowly towards the equator, they will there
be very much crowded together, and the figures of the
countries as they approach the primitive circle, will be very
much contracted in latitude. Scholium.
307. In the orthographic projection of a map, upon the
plane of a meridian, we have seen that the meridians are
ellipses ; but that, on the plane of (the equator, they are straight
lines; — that, in the former case, the parallels of .latitude,
are all straight lines ; but in the latter, they are concentric
circles; consequently, by that method the longitude, and by
this the latitude, are very much contracted towards the cir
cumference of the map. The contraction of the extreme
paits of the map, therefore, with the objections before stated
(125), render this projection very unfit for the construction
of maps.

95

Of the stereographic construction of Maps, first, upon the
Plane of a Meridian ; secondly, upon the Plane of the
Equator; and thirdly, upon the Horizon of a place.
308. Setting aside the oblate form of our globe, and con
sidering it as spherical, it may be perceived that the whole of
the visual rays extended to all the points of any circle formed
on the globe, constitute a cone, of which the section, in the
plane of the picture, can only be one of the curves of the
second degree, and even in some cases a straight line. It
would appear that the first decisions, in the choice of the
point of view, were dictated by the consideration of the
consequent facility in the construction of the map ; and that
from the time of Ptolemy, it had been Observed, that in
making the plane or picture pass by the centre of the sphere,
and placing the point of view at the extremity of the radius,
drawn perpendicularly on that plane, all these circles of the
globe were represented by other circles, of which the con
struction was easy, and which intersected each other on the
map, under the same angles as upon the sphere, so that the
rectangular quadrilaterals, comprehended between the meri
dians and parallels, -were represented by curvilinear quadri
laterals, also rectangular *. It has since been proved, that
the infinitely small portions of the globe, assume in this pro*
jection their natural figure; but it must be observed, that
this similitude only takes place in very small spaces. Such
are the conventions which have given rise to the stereographic^
projection, and such are its principal properties %.

* Ptolomaei Planisphserium, etc. Aldus Venetiis, 1558.
f The word is -derived from the 'Greek, meaning the art of drawing
the form of solids on a plane.
X La Croix's Intro, to the Paris Edition of Pinkerton's <Geog.

96

PROBLEM III.

To project planispheres for a map of the world stereogram
phically on the plane of a meridian, in the position of a right
sphere. See Fig. 7th, Plate XII.
309. The poles Pand^J, are now supposed to coincide
with the wooden horizon ; your eye is supposed to be on
the surface of the sphere, directly above the point C ; and,
therefore, PEpQ, would represent the glass plane we have
already (131) imagined: and, upon this plane, would the
meridians and parallels be drawn for one hemisphere. The
projection on paper would be identical to that on a glass
plane of the same diameter, and as p and P represent the
poles, and that E Q represents the equator, the point E may,
therefore, be the first meridian, or that from which the lon
gitude is reckoned. Now the meridians are projected into
circles, the radii of which are their distances from PE (136).
310. If you were required to draw the meridian, the lon
gitude of which is sixty degrees, the radius P O is to be con
sidered as unity, the secant of 60° is two, and therefore, with
an extent of your compasses equal to 2, set one foot in P,
and extend the other to E Q, which, in this case, you would
produce ; and with that point as a centre, sweep the circular
arc Pdp, which will be the real projection of that meridian.
In the same manner all the required meridians are to be
described. The next thing to be done, is to describe the circles of lati'
lude, whose radii will be the cotangents of the latitude.
311. If you were required to draw the circle of latitude,
representing sixty degrees, you would take E a equal to 60°,
and with an extent of compasses equal to -577, the cotangent
of sixty degrees, set one foot in a, and extend the other to

97
Pp, produced iF required, and, with that point as a centre*
describe the circular arc a c b, which will represent the pro
jection of that parallel*. I need hardly mention to you
now, that in the same manner all the- other parallels are to
be described. Compare this method with Problems VIII.
and IX. Section IV. and adopt which ever seems most
agreeable to your inclination, and most easy of execution.
309. Provided you describe the other circle BUCQ, of
the same dimensions as PQpE, and touching the latter in
Q, you may with one extent of compasses, always describe
four arcs of a circle before you shift the instrument, as was
observed to you in Section IV. article 168.
310. We have already shewn that the quadrilaterals com
prised between two consecutive meridians and parallels,
augment in extent in proceeding from the centre to the cir
cumference, — that this enlargement results from the obliquity
of the visual rays when they depart from that which is per
pendicular to the picture, and which may be called the
optical axis. See articles 151, 152, and 153.
PROBLEM IV.
To project the planispheres stereographically for a map
oftheworld, on the plane of the equator, in the position of a
parallel sphere. See Fig. 8th, Plate XII.
311. When the globe is placed in the position of a paral
lel sphere, your eye would be at P, directly upon its surface,
and A E Q U would represent the pellucid plane we have
imagined (131), on which both the meridians and parallels
would appear ; and as all the meridians would diverge from

f Dr. Vince's Astro. Art. Stereog. Const, of Maps.
K

98
your eye to the circumference of the plane of projection, they
would form so many radii ; their projection would then be
the straight lines drawn from P to the circumference E Q U A,
because, moreover, they are all perpend icular to that plane.
315. The parallels of latitude as they depart from the
pole towards the plane of projection, would seem to your eye
as so many concentric circles, whose common centre would be
P; and their radii would be the tangents of half the distances
of (he parallels from the pole.
- 316. And because the spiial rhumbs on the globe become
propenional spirals in the stereographic projection on the
plane of the equator, the rhumb line ab in this projection,
is manifestly the logarithmic spiral, as it is also in the equa
torial orthographic projection.
317. Here again the stereographic equatorial projection,
presents you with the parallels of latitude crowded together
about the pole ; but you gain one advantage, if such it may
be called, from this decrement at the poles: namely, an
increment towards the primitive circle, by which means,
those portions of the earth's surface near the boundary of
your map, may be tolerably fairly delineated.
Of the stereographic projection upon the plane of the
horizon of any place. See Fig. 9th, Plate XIL
318. We are enabled to fix the globe in any oblique pdsfc
tion, by means of the brazen meridian. This brass circle
serves not only the place of all the meridians, but, by being
graduated by nineties on each quadrant, we can elevate
either pole to any given altitude, and have thence a given
obliquity of the sphere. It is obvious, that the brass circle,
which represents the plane of any celestial meridian what
ever, and the wooden horizon, are the chief and only parts'
of the apparatus to be considered in horizontal projections;
for, as the axis of the globe admits of every possible incliaa-

99
tion with respect to the wooden horizon, this circle maybe
made to represent universally the horizon of any place.
316. Maps constructed on the plane of the meridian,
separate the adjacent parts of the globej and offer only in an
exact manner the respective situation and configuration of
the countries near the middle of the map. The polar pro*
jection remedies this defect, and represents the hemispheres
separated by the equator, displaying thereby with sufficient
exactness the regions around the poles. The horizontal pro
jection remedies both the foregoing defects, as far as regards
any particular place, as the representation of the hemispheres
above and below the horizon of the place to which they
tefer, are the most proper for the knowledge of the surround
ing regions or their antipodes. This projection, then, merits'
particular attention, as the oblique position of the sphere is
the most general to the inhabitants of (he globe.
320. In all horizontal projections, you are to consider the
place you are in as the centre, from which the design of tins'
projection is to show the distance and situation of all other
places. The position of the sphere in every horizontal pro
jection, is that of an oblique sphere. The extreme circle
will be the horizon ; the eye will be posited in the nadir, or
point diametrically opposite to the zenith of the. place iri
the centre; the plane of projection, or diaphanous medium,
will be coincident with the plane of the wooden horizon,
butting the globe into two equal hemispheres through the
centre, and at right angles with that fictitious line drawn
from the eye to the opposite point or place where yoii are*
321. Thiisj suppose you "were required 10 make a pro
jection of .this kind on the horizon of London*, provided
you had a glass sphere hung like the terrestrial globe, the

* Note O.
H 2

100
north pole being elevated to the latitude Of 51!", London"
would be in the zenith, and your eye would be in the nadir }
and if a translucid glass plane, coinciding with the wooden
horizon, were fixed to receive the picture, it is very obviousj
that rays falling from all places of the convex surface of the
sphere to your eye below, would pass through the imaginary
diaphanous plane or medium, and on that plane would be
projected every line, circle, and place, above the horizon,
but subject to these limitations ; those portions of the earth's
surface about the place on which the projection is made,
would be contracted, whilst those places towards the plane
of projection, would be enlarged.
322. The other half of this projection would be effected
by transferring your eye to the zenith, or London, when every
object beneath the horizon, would be represented on the
diaphanous plane, but subject to the limitations just refer
red to.
323. In the diagram referred to above, the pole is pro*
jected within the primitive circle, or plane of projection ;
and its distance OP, from the centre O, is equal to the
tangent of half the complement of latitude. The manner of
drawing the meridians will be found in the two following
Problems.

PROBLEM V.
To construct a map stereographically, for a horizontal
projection of the sphere, answering to the latitude of London.
See Plate XIII. Fig. 1st.
First, to draw the circles of latitude.
324. Assume on your paper any point, as Z, for the cen
tre of your map, and the situation of London ; and from Z
as a centre, with any radius, describe a circle WNES;

101
draw WE and NS at risfht angles to each other: NS will
be the first meridian, and WE will be the prime vertical,
or azimuth of east and west. (See article 356.) Divide the
quadrant EN, into nine equal parts, and at 51| degrees from
N, set off the point A, the elevation of the north pole for
London. From W to this point, draw the line W P A ; and
the intersection of this line with NS, as in P, will be the
place of the Arctic Pole.
325. Take the chord of any of the quadrants, as N E, for
instance, and from A, set off the space A B = N E. Draw
the straight line W B, and the intersection of this line with
N S, as at Q, will be the intersection of the equator with the
first meridian.
326. With the same extent of compasses by which you
divided the quadrant NE into nine equal parts, begin at B,
and set off three spaces towards S ; begin again at B, and
go entirely round to W, setting off equal spaces; in all,
there are twenty-four of these spaces to be set off. Then,
from W, to each of these points in the arc W N A B S, draw
straight lines, and the points of intersection of these lines on
the line S N produced, will give you the diameters of as
many parallels of latitude as are necessary ; that is, of twelve,
for there are no more visible above the horizon. Exclusively
ofWPA, seventeen of those straight lines drawn obliquely
from W, fall within the circumference of the primitive
circle; and their intersections with S N are, therefore, with
in the periphery of the projection ; but there are seven of
these oblique lines that intersect SN produced beyond N.
327. As you have now the diameters of your parallels of
latitude, bisect each diameter, because the parallels or circles
of latitude have not a common centre, and describe the
twelve circles of latitude through the points of equal degrees;
as 80, 70, 60, &c. the centre of 80 being the pole P, but
that of all the others in a point beyond P and N.
328. For the tropics and polar circles, set off the proper

102
quantity of each on the pr'mitive circle, draw straight lines
to determine the dismeter, as in the latitudes; bisect each
diameter, and describe those circles as they appear by the
dotted lines. Secondly, to draw the meridians.
329. Through the three points W, P, E, describe the
circle CPD; this last circle will be the plane of projection
for a stereographic planisphere on the plane of a meridian.
330. Delineate on this plane the meridians for a stereo
graphic projection by Prob. VIII. Section IV. and Plate VI.
But observe, in drawing these meridians, that you must
allow each of them to pass through the pole P, till it comes
in contact with the semi-circle WNE, otherwise your map
will be deficient of those meridians that stretch beyond the
pole. Thus, when you describe the meridian F P, continue
the sweep of your compasses till you arrive at G. The
meridian FI P, you would likewise continue, till you have
advanced to the point between G and A; and the same is to
be understood of all (he others.
331. At the same time that you are about one horizontal
projeciion, you may be carrying on two; and, by a little
attention and assiduity, you may make in a short time very
beautiful horizontal projections; whichj when tolerably well
cxeruted, are as elegant as is either the meridional or equa
torial projections ; but subject to this disadvantage, that you
have not that fine division between the new and the old
world, which the projection on a meridian exhibits.
332. But the meridional projection being properly the
basis of the horizontal, I shall, therefore, give you another
method, which is not only more easy than the last, but more
agreeable to the principles of the projection, and this will
be raising the superstructure upon the foundation.

103

PROBLEM VI.
To project planispheres for a map of the world stereo-
graphically upon the plane of the horizon, for the latitude
of London, by describing the meridional projection first.
Fig. 1st, Plate XIII.
First, to project the meridians.
333. The primitive circle in this case will be CPD.
Draw CD and PS (produced) at right angles, and divide
each quadrant into nine equal parts. Now, if C D represent
the equator, PS will represent the first meridian in the
horizontal projection, or the north and south azimuth (356).
334. From D to E, in the quadrant P D, and from C to
W, in the quadrant CP, set off 51° SCC for the latitude of
London, and draw the line W E, which will be the east and
west azimuth of the place. The point of intersection of the
line WE, with the first meridian N PS, represents London,
marked Z ; from which, as the centre of projection, with
the radius ZE, describe the plane of projection NESW,
which will represent the horizon of London.' The pole P,
in the meridional projection, is the pole in the horizontal %
and the arc of the primitive circle CPD, passing through
the pole P, and the points W, E, divides that meridian
distant ninety degrees from the meridian of London NZ S.
335. Describe the meridians on the meridional projection,
by Prob. VIII. Sect. IV. observing to draw them through
the pole, beyond the primitive circle CPD, to touch the
horizontal circle NESW; as you may more clearly appre*
hend by looking over the diagram.
Secondly, to project the parallels of latitude.
336. From W, draw oblique diverging lines through
every tenth degree in the meridian PDjdown to the point 30,

104
i. e. towards figure second. The points of intersection made
by these diverging lines on the meridian NS of the hori
zontal circle, are the points through which the parallels
must pass on this side of the pole; and because those paral
lels have not a common centre, in order to find the points
through which they are to pass on the other side of the
pole, produce SN indefinitely, and keeping still your ruler
on the point W, draw lines through every tenth degree of
the meridian PC. For example, the first beyond the pole
passes through 80 in P C, the second through 70 in PC,
and so on for the rest ; then, the points of intersection in the
first meridian NS, will be the opposite points through which
the parallels are to pass on the other side of the pole.
337. You may now mark the points on either side of P,
with the degrees of latitude 80, 70, 60, &c. on one side, and
with the same on the other.
338. Describe circles through the corresponding points,
or those marked with figures of the same denomination, for
the parallels of latitude. Thus, for 80, set one foot of your
compasses in P, and sweep a circle with the radius P80;
then bisect the line 70 70, and set one foot of your compasses
in the point of bisection, extend the other to either -70 for
a radius, and sweep the parallel of 70 degrees. Proceed
with the remaining parallels in this way, bisecting each
diameter or space between the corresponding degrees, to find
the centre, as the successive parallels have different centres.
339. By inspecting the diagram, you will find that the
points in the diameter NS, are always of the same denomina
tion with those by which they are made in the arc of the
meridian WPD. Thus, the parallel of 80 in PED, cuts
the parallel 80 in PS, the diameter; 70 cuts 70, &c. ; 50 in
PED, cuts 50 in PS; as 51° 30' in the quadrant PED,
cuts the latitude of London in the meridian PS: again,
beyond the pole 80, 70, 60,-&c. in the quadrant PC cross
the parallels 80, 70, 60, &c. in the meridian PN, or PN pro-

105
duced ; and finally, the point of the meridian cutting the equa^
tor in D, intersects the horizon in B, and the first meridian PS,
in Q, which is therefere the point through which the ambit of
the equator must be projected in the horizontal construction.
Thirdly, to complete your map.
340. Describe a circle round the horizon, to contain
the degrees of graduation. This graduation is not made
between the parallels, but must be made between the meri
dians, each of which is to be divided into 10 parts or de
grees, to show the longitude of places. The latitude must
be graduated on the first meridian NS, and numbered from
the equator towards either pole, and from the pole backward
towards N. Next, describe a circle without this last one,
wide enough to hold the figures belonging to the numbered
degrees. Lastly, describe two more circles, the first close
by the former, and the last sufficiently distant to admit of
the numbering of the rhumbs. Finally, divide each qua
drant of this last circle into eight equal parts, which will give
you, in the whole horizontal periphery, 32 divisions, being the
points of the mariner's compass, to show the bearing of all
places in the hemisphere, in respect of London the centre.
341. All the circles of the foregoing graduation, are seen
in Figure 2d, termed, Annexed Figure, in which so much
of the earth is described, as is contained within the horizon
of London. The diagram will serve you as a specimen of
the nature and use of this projection.
342. The outward graduated circles supply the place of
the azimuths, and whilst you are thence saved the trouble of
drawing them, the confusion they might occasion, supposing
they were drawn, is completely destroyed. When you
have thus completed your projection, insert all the places as
accurately as possible, agreeably to the directions already
given in Section II. and colour the different countries at

106
pleasure ; then, to finish the whole, fake a slip of Bath paper,
and make a central ruler of it, by graduating one edge with
the same parts into which Z N is divided. This ruler must
now be fixed by a rivet in the centre point, or in the locus
of London, and being moved about to any place, you may
presently tell the distance thereof; and the bearing or point
of the compass, on which it lies from London, may as easily
be known by the section of the outward or azimuth circles,
on which the points of the compass are represented.
343. At the same time that you are about one horizontal
projection, you may be carrying on two, from a meridional
projection ; "that is, one for the northern, and the other for
the southern hemisphere, provided the latitude of the place
in the centre does not exceed 55°, because then the horizons
would touch one another ; that is, the horizons of the hori
zontal projection would touch the equator in the meridional
projection, and consequently meet there.
Formulm connected with this Projection.
344. When you would know what size your meridian
projection must be, to project from it a horizontal projection
of any given dimensions, you will find the diameter of the
meridian projection by this proportion,
As cosine of latitude, is to radius;
So is the given semi-diameter of the horizontal projection,
To the semi-diameter of the meridional projection.
345. The semi-diameter of the meridional projection being
given, to know the scrai-diameter of the horizontal projection,
As radius, is to cosine latitude;
So is the semi-diameter of the meridional,
To the semi-diameter of the horizontal projection.
346. Having the semi-diameters of both projections given,
to find the latitude, say,

107
As the semi-diameter of the meridional projection,
Is to the semi-diameter of the horizontal projection;
So is radius, to cosine latitude.
347. And observe, that the versed sine of the co. lat. in
the meridional projection, is equal to the half tang, of the
comp. latitude in the horizontal projection, as has already
been shown.
348. When we wish to know the bearing and distance
only of places from the centre, the following method for
describing the horizontal projection may be adapted.
PROBLEM VII.
To draw the horizontal projection with azimuth lines, to
shorn the: bearing and distance of all places within the map
from London in the centre. See Fig. 1, Plate XIV.
349. Describe a circle, and cross it with two diameters
W E and N S ; the former is the east and west line, the latter
is the meridian of the place ; and the intersection of those
two lines at Z, will be the situation of London in the centre.
35Q- Divide each quadrant into 90 degrees, and in another
circle described within the former, draw radii for the 32
points of the compass; or, only the extreme portions of those
radii, as they appear within the inner and central circles
of the lower figure, marked N, NE, E, &c. \
351. This done, you may put in the centre London,
Paris, Amsterdam, Constantinople, or any other place from
which the distance and bearing of others are to be determined.
Then, by the help of tables of bearing and distance, you may
lay down all places according to their respective situations.
You will observe, that when in geography or navigation,
bearing is spoken of, the position of one place with regard
to another, as estimated by the points of the compass, is
meant. Thus, a place is said to bear N. E. or N. N. E. &c.

108
when it lies from your present situation in either of those
directions. And distance in geography, if we use the word
according to its proper acceptation, is the arc of a great
circle intercepted between two places; or, according to its
common acceptation, it is the space considered barely in
length, between any two places— it is the shortest line that
can be drawn between them, and is thence sometimes called
the crow's flight measure. In navigation, distance is the
number of miles or leagues that a ship has sailed frpm one
point to another T
352. In general, it may be observed, that the distance be
tween any two points on the globe, is measured by bringing
one of them to the meridian, and placing the centre on
whioh the quadrant of altitude turns directly over it; then
turning the quadrant of altitude round till it passes through
the other point, and the degrees intercepted on the arc will
be the distance required.
353. The azimuth is an arc of the horizon intercepted be
tween the meridian of (he place, and the vertical circle pass?
ing through the centre of an object, and is equal to an angle
at the zenith, which is measured by the forementioned circle.
The azimuth is reckoned eastward in the morning, apd west
ward in the afternoon; and is usually estimated to the north
or south, as it is nearer to pne or other of those points: thus,
if it be found that the vertical circle which passes through
the zenith, intersects the horizon exactly between the E. and
S. then the azimuth of the object is said to be 45° eastward
of the south ; or it is the complement of the eastern and western
amplitude. And the amplitude is an arc of the horizon, in
tercepted between the true east and west points, and the centre
of the sun or star at its rising or setting -. it is, therefore, of
two kinds, ortive or eastern, and occiduous or western. These
are also called northern or southern, as they fall in the north
ern or southern quarters of the horizon; and the complement
of the amplitude, or the distance of the point of rising or set-

109
ting from the north or south point of the horizon, is called the
azimuth. Azimuth circles, called also vertical circles, are,
therefore, great circles of the sphere, intersecting each other in
the zenith and nadir, and cutting the horizon at right angles*.
354. If the direction or azimuth which one point makes
with the other, be required, one of them must be brought
into the zenith or pole of the horizon; that is, the globe
must be rectified for that point, or the pole elevated to the
latitude of the place, and the horizon of the globe will then
represent the horizon of the place: this being done, and the
quadrant of altitude adjusted in the manner above described,
the number of degree^ intercepted on the horizon by the qua
drant of altitude, and the north or south point of the horizon,
is the azimuth required, which is the angle that a great circle
passing through the given points makes with the meridian.
355. When, therefore, you have, on a horizontal projec
tion for London, to lay down places from the globe, or maps,
you have only in the former case to bring London to the
meridian, and elevate the pole according to its latitude
51° 30'; fix the quadrant of altitude in the zenith, and
apply it to every place whose situation you would represent;
for example, to the principal cities in Europe, to the sea-
coasts and boundaries of the countries you would lay down;
and observe the angles which the quadrant makes with the
meridian in each application, that is, the angles of position.
from London ; at the same time observe how many degrees
ef the quadrant lie between London and the place, and that
will be the distance. At each application of the quadrant,
you. may find the distance of many places, by marking at
what degree the edge of it passes over them; and then they
have the same angle of position.
356., The angle of position between two places, is an angld
art the zenith of one of the places, formed between the brazen
«*— ,  ,  ¦»¦ ;  ¦ '   — ¦ ¦¦'- ¦ ' ' -
.* Note P.

110
meridian and the quadrant of altitude passing through thj$
other place, and is measured on the horizon.
357. For transferring into your map the bearing and dis
tances of places so taken, if the representation, do not exceed
two or three kingdoms, you may have a scale of equal parts,
that shall extend from the centre to the horizon. This scale
is seen on the upper circle of Figure 1st, but it must be
moveable, and rivetted on the map. This central ruler being
fixed on London in the centre, may now be moved to the
degrees in the liorizon, according to the angles of position of
the places to be laid down ; and close by the edge of the
ruler, the degree and minute by which any place is distant
from London, is to be carefully marked. Thus you may
proceed to insert as many places as you want to put down in
your map, with respect to the bearing and distance from that
city. But should your map be a whole planisphere, then this
scale must be graduated according to the projection and ele
vation of the pole.
358; The inequality of the spaces of the graduation of the
stereograpbic projection does not, in general, permit the
application of a rectilinear scale, to compare the respective
distances of places, distances which are measured according^
to an arc of the great circle, which joins these places two
and two; but we may always, by means of the graduation
itself, measure the distance between the centre of the map*
and any one of its points; and we may, in consequence, find
upon a horizontal projection, referred -.to London, for exam
ple, the distance from this city to alt the other parts of thet
globe; This property is the consequence of a projection,, iif
which all tlie great circles which pass by the. centre of the
map, intersecting each other according to the optical axisy
bave for their perspectives right lines drawn by that centre,
and admit a graduation similar to that which is marked upon
the equator of maps. o£ the world constructed upon the plane.
of the meridian.

PZATE XJV.
page, Hi?

A. taw's.?'' n dd.

Lowrif sculp.

in
SECTION VIII.

ON THE PRINCIPLES AND PRACTICE OF DEVELOPING A
SPHERIC SURFACE ON A PLANE*.
359. X HOUGH every kind of projection furnishes the
means of ascertaining the distances and angles concerned in
that projection, and though the representation of the whole
surface of the sphere by means of a projection, is the most
eligible of all others; yet, when a small portion is required
to be represented, the method of developing the surface is
preferable, not only on account of the developeraent being
more nearly similar to the spheric surface Itself, but also on
account of the distances being obtained by inspection only,
or from a scale of equal parts ; and though this mensuration
is only an approximation to the truth, it is sufficient for
common use. Granting the impossibility of forming a true
developement of the spheric surface, if, however^ the sphere
be supposed to be circumscribed by a polyhedron, the' side
of the polyhedron extended upon a plane, will be xery nearly
a true developement of the surface.
360. In approximating the developement of the spheric
surface, there are two different methods of considering the
subject; one is, by supposing the polyhedrous surface to
Consist, of those conic frustrums, each touching the surface
of the sphere in the middle ; and the other is, by supposing
the polyhedrous surface to be composed of those of a cylin
der, meeting each other in planes passing through the centre
of the sphere, and intersecting each other in one common
diameter as an axis; and because both conic and cylindrical
* NoteQ.

112
surfaces are developable, the surface of the circumscribing:
polyhedron will also be developable, and therefore may be
correctly represented.
361. Now, supposing all the sides to be equal, and increas
ed in number at pleasure, the greater the number, the nearer
will the surface of the polyhedron coincide with that of the
sphere ; and consequently, any portion of the surface of the
developement of the surface of tlie polyhedron, will be very
nearly equal and similar to the corresponding portion of the
spheric surface.
362. Each of these different methods of representing the
spheric surface, has its peculiar advantage and disadvantage,
when applied to the art of map-making: the conic method
of developement is best adapted to represent countries to any
extent in the difference (27) of longitude, or round the whole
circumference (28) of the earth if required, to a certain extent
in difference of latitude (24); and the cylindric method of
developement is well adapted to represent countries contained
between any two parallels of latitude, but not to have any con
siderable difference of longitude.
363. Of these two different projections by developement,
the conical projection is the most simple, and is attended
with the least trouble, on account of the facility of describing
the parallels of latitude in concentric circles, and the meri
dians in straight lines.
364. In comparing a spherical zone to a truncated cone,
in order to construct its developement, we view the parallels
as circles described from the summit of the cone taken as a
common centre ; and the meridians are right lines subjected
to pass through that point. It is visible that the result will
approach the nearer, in proportion as the map shall embrace
less extent in latitude.
365. This projection may vary in different ways; for it
may be supposed that the cone is a tangent to the middle
parallel of the map, and in consequence, exterior 3 or7 that

113
it may be in part inscribed in the sphere, that is to say, form
ed by the secants of the meridians. In the first case, the
map will not be perfectly exact, except on the middle paral
lel, which will preserve in its-developemeat the length wliich
it really possesses on the sphere; but the parallels placed
above and beneath, will exceed those on the sphere, which
are correspondent.

APPLICATION OF THE DEVELOPEMENT OF THE CONIC SUR
FACE IN THE CONSTRUCTION OF MAPS.
366. With respect to the application of the developement
of the conic surface in the construction of maps, the Rev.
Patrick Murdoch proposed to substitute to the tangent cone,
a cone partly inscribed and determined by this condition,
that the part of the area comprehended in the map, should
be equivalent to that of the spherical zone which it repre
sents*. 367. Though it may be difficult, at first sight, to conceive
that the surface of the globe can be represented by a part of
the surface of a cone, yet this is the case in regard to the
map under consideration; for, you may plainly see, that
the form of Fig. 3d, Plate XIII. bespeaks it to be a part of
the surface of a cone. You may easily form the surface of
a cone, of any plain piece of paper cut into a circular form
or base ; and you will as easily consider, that if a cone about
twice the height of the semi-diameter of the globe, were to
be conceived as standing on the same base with the hemi
sphere, namely, on the equator, the surface of such a cone
Would in part lie within the surface of the globe ; and then,
nothing can be more natural, than to suppose that the sur-

* Philos. Trans. Vol. L. Pait II. for 1758, p. 553.
J

114
face of the globe at so small a distance from the surface of
the cone, might be very easily projected or delineated on it ;
and in such a case, the projection of the countries and their
bearings, distances, &c. will be very nearly the same oh the
surface of the included part of the cone, as on that of the
globe itself; and when such a geographical conical surface
is cut out and expanded, it makes the map now under
consideration. 368. Mr. Murdoch observes, that when any portion of the
earth's surface is projected on a plane, or transferred to it
by whatever method of description, the real dimensions, and
often the figure and position of countries, are much altered
and misrepresented. The particular methods of description
used among geographers are so various, that we might sus
pect them to be faulty ; but in most of their works we find
these two blemishes, the linear dimensions visibly false, and
the intersections of the circles oblique; so that a quadri
lateral space shall often be represented by an oblique-angled
rhomboid figure, whose diagonals are very far from equal;
and yet, by a strange contradiction, you shall see a scale of
distances inserted in such a map.
369. The last of these blemishes is removed, and the other
lessened, in some maps of P. Schenk's of Amsterdam, a
map of the Russian Empire, the Germania Critica of the
famous Professor Meyer, and a few more*. In these the
meridians are straight lines,, converging t© a point, from
which as a aentre, the parallels of latitude are described,
and a rule has been published far the drawing such mapst.
But as that rule appears to be only an easy and convenient
approximation, it remains still to be inquired,
JVhatis the construction of a particular map that shall

* Senex drew several maps of that form.
t See the Preface of the small Berlin Atlas.

115
exhibit the superficial and linear measures in their truest
proportions? (366.) In order to which,
370. LetE/LP,Fig. 3d, Plate XIII. be the quadrant
of a meridian of a given sphere, whose centre is C, and its
pole P ; EL, E/, the latitudes of two places in that meri
dian, E M their middle latitude. Draw LN, ?s cosines of
the latitudes, the sine of the middle latitude M F, and its
cotangent MT. Then, writing unity for the radius, if in
Nra
C M we take C x = L/ x MF x MT' and tlirouSa x we
draw xR, xr, equal each to half the arc LZ, and perpen
dicular to C M ; the conical surface generated by the line
Rr, while the figure revolves on the axis (i. e on Ct) of the
sphere, will be equal to the surface of the zone that is to be
described in tlie same time by the arc L/; as will easily ap
pear by comparing that conical surface with the zone as
measured by Archimedes.
371. And, lastly, if, from the point t, in which rR pro
duced meets the axis, we take the angle Ct V, in proportion
lo the longitude of the proposed map, as M F the sine of the
middle latitude is to the radius, and draw the parallels and
meridians as in the figure, the whole space SO Q V, will be
the proposed part of the conical surface expanded into a
plane ; in which the places may now be inserted according
to their known longitudes and latitudes,
372. To form the degrees of longitude, we must take the
360th part of an arc, described from the summit t as the
centre, with the radius t r, and which represents the develope
ment of the parallel passing by /and S, then drawing right
lines through the divisions of the arc SV, and t the summit
of the cone, we shall have the meridians. In order to pro-
Cure the degrees of latitude, we must bear upon one of those
meridians, beginning at the point M, as well above as be
neath, parts equal to the developement of the arcs on tlie
i2

116
terrestrial meridian. In fine, we describe from the point P,
and, by the divisions of the meridian P E, concentric circles
which will represent the parallels!. Thus, these concentric
circles, having t the vertex of the cone for their common cen
tre, take as radii tr = tS oxtlO, ty = t20,tz = t50j
tR = t60.
EXAMPLE.
Fig. 3d, Plate XIII.
Let LI bd the breadth of the zone, lying between 10" and
60° north latitude; let its longitude be 110°, from 20° east
of the Canaries, to the centre of the western hemisphere;
comprehending the western parts of North America, and the
ocean that separates it from the old continent.
i Nw
- 373. And because Cx = ,, »p  jjTm , add these
three logarithms ; and take their sum from the log. N n, the
remainder is the logarithm of C x.
374. And because 1 : Cx : : ME : xt, to this adding the
log. M T, the sum is the log. of xt; and x R, (= x r — J
hl,)Rt will be found, as will also r t.
375. Whence, having fixed upon any convenient size for
our map, the centre t is easily found. As, allowing an inch to
a degree of a great circle, or 50 inches to the line Rr, Rt,
the semi-diameter of the least parallel, will be 54'255 inches,
and that of the greatest parallel 104"255 inches.
376. Again, making as radius is to MF, so is the longi
tude 110°, to the angle Si V; that angle will be 63° 5'| .
Divide the meridians and parallels, and finish the map as
usual. 377. Observe, that the log. MT being repeated in this
computation with a contrary sign, we may find xt imme
diately by subtracting the sum of the logarithms nf LI and
M F, from the log. of N n.

117
378. A map drawn by this rule, will have the following
properties :
379. First, The intersections of the meridians and paral
lels will be rectangular, as on the surface of the globe.
380. Secondly. The distances north and south will be
exact ; and any meridian will serve as a scale to measure
these distances by.
381. Thirdly. As such a cone on which this map is
made, is supposed to pass through the surface of the globe in ,
two places, the parallels of latitude in those places where the
cone intersects the globe, will be the same in the map as on
the globe itself; that is, all distances east and west may as
truly be. measured on them. For example: The parallels
through z, or 50° latitude, and y, or 20° latitude, where the
line Rr cuts the arc LI, or any small distances of places that
lie in those parallels, will be of their just quantity. At the
extreme latitudes they will exceed, and in mean latitudes
from x towards z or y, they will fall short of it ; but, unless
the zone is very broad, neither the excess nor the defect will
be any where very considerable,
382. For as the middle part of the map on the surface
of the cone, lies within the globe, the meridian on that
part of the globe will, therefore, be at a greater distance
than those parts of the meridians projected on the map; and,
consequently, the parallel of 35°, and those pear it, will be
deficient from the globe, — they will give a less distance than
what is just; — any two places on the middle of such a map,
must be represented something nearer together, than they are
upon the globe, if their longitudes be expressed exactly.
383. On the other hand, those parallels which terminate
the map, have an error in excess ; thus, the parallels of 10»
and 60°, as they are projected from parts of the globe which
lie within the cone, must have a greater space between the
meridians, than their corresponding parallels on the globe
itself; and thus, places in those parts of the map, are repren
Rented at too great a distance from - ' JU r.

118
384. Fourthly. The latitudes and the superficies of the
map being exact by the construction, it follows, that the
excesses and defects of distance now mentioned, compensate
each other ; and are, in general, of the least quantity they
can have in the map designed.
385. Fifthly. If a thread is extended on a plane, and
fixed to it at its two extremities, and afterwards the plane is
formed into a pyramidal or conical surface, it may be easily
shewn that the thread will pass through the same points of
the surface as before; and that, conversely, the shortest dis
tance between two points in "a conical surface, is the right
line which joins them, when that surface is expanded into
a plane. Now, in the present case, the shortest distances
on the conical surface will be, if not equal, always nearly
equal, to the correspondent distances on the sphere; and,
therefore, all rectilinear distances on the map, applied to
the meridian as a scale, will nearly, at least, shew the true
distances of the places represented.
386. Sixthly. Another, and not the least considerable
property of this map is, that it may, without any. sensible
error, be used as a sea chart; the rhumb lines on it being
logarithmic spirals to their common pole t, as is partly re
presented in the diagram ; and the arithmetical solutions
thence derived, will be found as accurate as is necessary in
the art of sailing.

387. If it is required to draw a map, in which the super
ficies of a given zone shall be equal to the zone on the sphere;
while, at the same time, the projection from the centre is
strictly geometrical;
388. Take Cx to C M, as a geometrical mean between
C M and N n, is to the like mean between the cosine of the
middle latitude, and twice the taDgent of the semi-difference
of latitudes; and project on the conic surface generated by

119
xt. But here the degrees of latitude towards the middle,
will fall short of their just quantity, and at the extremities
exceed it, which hurts the eye. You may use either rule,
or, in most cases, you need only make C x to C M, as the arc
ML is to its tangent, and finish the map; either by a pro
jection, or, as in the first method, by dividing that part of
x t which is intercepted by the secants through L and /, into
equal degrees of latitude.
389. Mr. Martin, in the second volume of his " Young
Lady and Gentleman's Philosophy," has shown a rule de
rived from a well-known principle of the centre of gravity,
in order to obtain the distance C #;T-the formula is, C x =
CM xLB ,,, . ., , ,. T .
- — -jt-j-j  , which gives this analogy ; as any arc ML is
to its sine LB, so is radius C M to C x, the distance of the
centre of gravity x of twice that arc LI, which is very con
venient in practice. In the same volume he also gives an
example as applied to a map ; the following is the rule :
390. In this new map (Murdoch's) the extent or differ
ence of latitude is 50° = LI; therefore, ML=? 25° =
2*6179 inches to a radius of 6 inches*. Also, the sine BL
of the same arc is 2 "5358 inches, therefore, say,
As the arc ML = 26179  0-417953
Is to its sine BL = 2-5358  0-404099
So is radius CM = 6-  0-778151 1-182250
To the distance C x = 5-8116  0-764297
So that the side pf the cone Qt, falls within the zone LM/
about jth of an inch s=? M#j which is the versed sine of
14* 24' = M z or My.

* See Martin's Inst. &c, of the NewtQni*yi ftfathasjs, vol. i. page 3p2,
art. 885, Doct. Fluxions.

120
391. The advantage gained by Mr. Murdoch's method of*
constructing maps, is only in facility of description; for,
though the whole area is truly represented, yet the measures
taken along with the parallels of latitude are very incorrect,
being no where equal to those of the sphere, except where
the conic surface intersects that of the latter body. The,
conic meridians within the intercepted part, are less than
the true meridians, and those on each side of the intercepted
part greater. This method might, therefore, be restricted
not to exceed 30° of difference of latitude, and in this case
it would be much simpler to take the surface of the cone a
tangent to the sphere in the parallel circle of middle la
titude. 392. From the rule which Mr. Murdoch gives for finding
the vertical angle C / V, may be derived that of finding the
vertical angle of developement, in order to cover any portion
of the surface of a cone, the side of the cone, the radius of
the base, and the degrees contained in the circumference of
the base being given. Mr. Murdoch's rule is,
As radius,
Is to the sine of middle latitude ;
So is the difference of longitude,
To the vertical angle Ot V. ; 
but radius is to the sine of middle latitude, as the cotangent
is to the cosine; therefore, as cotang. : cosine of mid, lat.
: : diff. of long. : vert. 4CtY- Now the cone that would
touch the sphere, has its side equal to the cotangent, and the,
jradius of its base equal to the cosine ; therefore,
As the radius of the cone,
Is to the radius of the base;
So is the number of degrees contained in the circura-»
ference of the base,
To the vertical angle of developement.
S93. The astronomer Dellile (de la Croyere), who was
phargcd with the construction of a general map pf tfc^

121
Russian Empire, chose the conical projection; but, in order
to perfect it, he thought of making the cone enter into the
sphere in such a way, that it should intersect it according to
two parallels, each placed at an equal distance from the
middle parallel, and from one of the extreme parallels.
394. Eulcr occupied himself with this projection, but he
substituted to the determination of the parallels, which must
be common with the sphere, that of the point of concourse of
right lines, which represent the meridians, and of the angle
which they make among themselves in the comprehended.
degree of longitude*.
To construct a Map on the Principles of Developement.
395. First. Suppose that APR (Fig. 2 No. 1 , Plate XV.)
is one half of a sphere or globe, of 1 inch and Aths in dia
meter, and that we are to develope one-half of this half sphere
on a plane surface ; the developement would be one-half of
the whole sphere or globe, of which APR is a half. But
were this half sphere cut in two, APR would be its plane
of projection, being the plane of that meridian in which the
semi-sphere was bisected, and we should then consider APR
merely as an arc of a great circle of the sphere; it would
be a semi-circle.
396. Secondly. Let, then, this semi-circle be bisected by
the perpendicular KP; let the quadrant AP, be divided
into nine equal parts in the points B, C, D, &c. ; let the
radii KB, K C, &c. be drawn ; and draw also the tangents
Bb,Cc, &c. to meet the perpendicular KP produced inde
finitely towards S.
397. Thirdly. From A as a centre, (Fig. 2, No. 2,
Plate XV.) with a division of the quadrant No. 1, set off
on No. 2 the equal spaces AB, BC, CD, &c. With the

* Acta Academiae Petropolitanas, torn. 1. pars, 1.

122
radius A P, No. 2, describe the semi-circle IPl, and tins
semi-circle shall be a developement of the semi-spheric sur
face APR; that is, IP I shall be a developement of the
fourth part of that sphere, of which APR is one-fourth
part. For, if we draw chords to the divisions or arcs,
AB, BC, &c. No. 1. the quadrantal arc AP, will then cir
cumscribe the fourth part of a polygon of thirty-six sides;
and if we transfer those chords to No. 2, by the distances
AB,BC,' &c. the whole line A P, No. 2, shall be equal to
the sum of all the chords of the arc of the quadrant AP,
No. 1 ; and, therefore, the semi-circle IP I, No. 2, shall be
a developement of the spheric surface APR, No. 1.
398. Fourthly. To draw the parallels in the semi-circle
No. 2, produce A P indefinitely beyond P, and on this line
transfer the tangents of latitude, that is, Bb, Cc, Drf, &c.
from No. 1, and ihey will be centres about which to describe
the parallels of No. 2. Thus, for example, the . tangent
Bb, No. 1, being the radius of the parallel that passes
through B, No. 2, you must in AP (No. 2) produced, set
off with your compasses a space equal to the tangent Bb,
No. 1 ; and from the point beyond P, No. 2, as a centre,
describe the parallel passing through B. Do the same by the
others, and all the parallels will be drawn.
399. Fifthly. To describe the meridians, divide Al, A I,
each into nine equal parts, do the same with each of the other
parallels, and draw curvilinear lines through the correspond
ing divisions of the parallels, so shall the meridians be drawn
for No. 2.
400. Figure 3, Plate XV. shews how a map of this
kind may be constructed, when you are not going to develops
any given spheric surface.
401. First. From O as a centre, with any radius OC,
describe the semi-circle A P C ; draw O P at right angles to
ACj divide the radii OA, OC,OP, each into nine equal
parts; through the points of division in OP, draw from the
point A, the diverging lines A 10, A 20, &c. and from O,

Fig.S.

D P

KgJ

.H
PF
sr
1
N
s
K I
¦/---
LK I
\ 7
\\
x\\
- '¦
"•-..<
^
1
S
(
T
A
Pag c 122.
Sojeso
l Rs
IMHpNI
Fig. 2. Nfl
123
draw the radii O 10, O 20, &c. to the points where those
diverging lines are terminated by the arc CP. Then, from
the extremities of the radii O 10, O20, &c. draw the tan
gents 10, 20, &c. meeting O P produced indefinitely; so shall
the points 10, 20, 30, &c. in O P produced, be centres, about
which to describe the parallels of latitude, according to their
respective radii.
402. Secondly. Having drawn these parallels, divide each
of them into nine equal parts, and through the corresponding
divisions of each parallel, draw curvilinear lines, which
will be the meridians required.
403. The other semi-circle done in a similar manner, will
complete the planisphere according to the method' of deve
lopement. But you will observe that, as the meridians are
not subject to any geometrical problem for describing a cir
cle, they must be drawn with a steady hand, in order to
pass through their respective points, and be, at the same
time, as curvilinear as their nature will admit.
SCHOLIUM.
404. This is a very near approximation to the truth — a
very near developement of the surface of the globe. All the
meridians and parallels intersect each other at right angles.
The successive quadrilaterals between any two meridians,
are nearly equal in area, and similar to those on the globe;
those adjoining the central meridian, are almost exactly the
same as their correspondents on the sphere. Every parallel
of latitude is equally divided by the meridians as they are on
the globe ; and every parallel, as terminated by two meri
dians on the map, is exceedingly near the same length as the
corresponding part on the globe itself. It must, however,
be observed? that the meridians are here represented of dif
ferent lengths, all longer than those on the globe, except A P
the central one. In this respect the method of develope
ment is the converse of every meridional projection of the

124
sphere, every meridional projection diminishing the meri*
dians as they approach the centre of the map; but even
in this respect it is less defective in the lengths of the meri
dians, than is Mr. Murdoch's with respect to the parallels of
latitude. However, since the central meridian answers as a
scale, the length of every meridian can be correctly ascer
tained; and the method of developement may, upon the
whole, be considered the most correct and the easiest of con
struction of any yet published. It may be employed in the
construction of the largest maps as well as the smallest, and
will be the more exact, in proportion as the difference of
longitude is less.
405. Some geographers have also entertained the idea of
developing in a right line all the parallels, and one of the meri
dians, that passing through thexmiddle of the map; thus, the
parallels, which are all perpendicular to this meridian, cor
respond in spaces with the globe. There are then assumed in
each of the. parallels, the degrees of longitude, according to
the laws of their decrease, that is to say, proportioned to the
cosines of the latitude. In fine, there passes through each
series of the corresponding points of the division, a curve
line, which represents the meridian. From this construction,
«f which Fig. 3d, Plate XVII. offers an example, it follows
that, in respect to its parallels, the map represents through
out, dimensions equal to those of the sphere; but the con
figuration is considerably altered on the sides, by the obliquity
of the meridians, so that the spherical rectangular quadrLj
laterals, comprised between the meridians and the parallels,
are represented by mixtilinear trapeziums, of which the
angles are very unequal, but the areas are in truth equal.
This projection has been employed in the Atlas Celesfis of
Flamstead; in 4 parts of the world, by J. B. Nolin; and,
by several other geographers *.

* Note R.

125

SECTION IX.

«EOMETRICAL PROJECTIONS OF CHOKOGIIAPHICAL MAPS
OF EUROPE, ASIA, AFRICA, THE AMERICAS, ENGLAND,
SPAIN, AND PART OF ITALY, THE RUSSIAN EMPIRE, ANO
A GENERAL METHOD OF DRAWING ANY PARTIAL MAPS*.
406. JL HE great utility of particular maps of countries,
arises from this, that they present us with a just view of the
several particular parts of the world, as they would appear
to an eye on the surface, or at a given distance from the
surface of globes so large as it is not in our power to have.
407. From particular maps accrues one considerable
advantage, of which we could by no other means become
possessed ; namely, that in such maps we can easily express
the true form and dimensions of countries, together with
the length of the several degrees of latitude, in the same
proportion as they lie on the surface of the earth itself; that
is, on the surface of a spheroid, for though we use a sphere
to represent the earth, it has long ceased to be doubtful,
whether the earth is a sphere or globe. Yet, as any globe
we are capable of making, will be much too small to shew ,
that difference which would be very sensible in globes of
20, 30, or 40 feet diameter, we slilLenjoy that advantage in
representations of the particular parts of such very large,
globes, in our common geographical maps of countries.
And were maps of countries taken from the surface of a

* Note S.

126
spheroid, and not of a sphere, the degrees of latitude in
them would not be precisely equal, as we find them in com
mon maps.
408. In this treatise, our circular maps are supposed to
be representations of a sphere; but you will find in the
Young Gentleman and Lady's Philosophy, under the title
Philosophical Geography, a set of maps constructed from a
spheroid of the same dimensions, wherein not only the degrees
are of an unequal length, but likewise the extent, situation,
and form of the countries are different. Such nicety would
not be necessary in the construction of maps, since the differ
ence between the polar and equatorial diameters of the earth,
does not exceed thirty-four English miles, according to the
calculations of Sir Isaac Newton, and other eminent astro
nomers. La Place concludes that the earth is an ellipsoid of
revolution, the equatorial and polar axis of which are in the
proportion of 578 to 577*; and Newton supposing the earth
a homogeneous fluid, found the ratio of the equatorial to the
polar axis to be 230 to 2S9 + . The difference referred to above
will give about the 230th or 235th part of the diameter. Now
let us transfer this to one of our globes, mounted for the solu
tion of problems, say a globe of 18 inches diameter, and it will
be found that the difference of the diameter will not exceed the
7£^ of an inch: now if this error is so small in the globe, it
must be much less in a partial map. Moreover, when we con-
skier the difficulty of executing globes large enough to shew the
details of geography, and the embarrassment occasioned by
their use, it will not be wonderful that we have been taught the
necessity of representing on a plane surface, the respective
situation of different objects on the globe of the earth; and as
cones and cylinders are capable of extension On a plane, with-

* System of the World, vol. ii. p. 119.
f Id. p. 121. See also chap. xii. vol', iu System of the World.

127
out rent or fold, while the curved surface of a sphere or
spheroid, is quite incapable of this extension, it must appear
evident that we are obliged to have recourse to different con
structions, in order to represent, at least in an approximate
manner, the natural relations between the extent of countries,
the distances of places, and the strict resemblance of con
figuration. 409. Perspective representations of the globe, or parts of
its surface, taken from different points of view, and upon.
different planes, and the species of developement, subject
to the laws of approximation, and confined to the relations
which are intended to be preserved, have been reduced by
Lambert, and after him by Euler and Lagrange, to the
general principle of the transformation of circular ordinates*,
assumed from the sphere; namely, meridians and parallels
into either straight or curved lines traced on a plane, and
depending upon conditions relative to the desired qualities
of the map f.
There may be employed in projecting partial maps, one
or other of the following methods :
1st. Project the meridians and parallels into straight lines.
2ndly. Project both the meridians and parallels into curve
lines. 3rdly. Project the parallels into curve lines, and the
meridians into straight lines.
4thly. Project the meridians into curve lines, md the
parallels into straight curves.

* Things arranged and dependent on the same order.
f La Croix's Math, aad Crit. Geo.

12$

PROBLEM I.
To draw a map of any particular portion of the earth 01/
method 3rd; as for example, a patt of Italy lying within
the 39th and 115th degrees of latitude. Fig. 1, Plate XVL
410. Draw a horizontal line E F, which bisect hi D, and
raise the perpendicular DC; then, because there are only
6° contained in this part of Italy we have chosen, you will
divide DC into six equal parts, through each of which a
parallel of latitude must pass ; and through C draw a line
parallel to EF.
411. Divide a degree into ten, or if large, into 60 equal
parts, by a diagonal scale : and from the table of geogra
phical miles, page 10, you will find the number of miles
which a degree of longitude contains in the latitude of 39 de
grees to be 46.62; and you would, accordingly, from your
scale, set off 23.31 (the half of 46.62) on each side of D. The
method of constructing a diagonal scale, may be known by
consulting Problems IX, X, and XL of this Section.
412. You will next find in the same table, the number of
miles contained in a degree of longitude, in the latitude of
45°, to be 42.43; and from your diagonal scale, take 21.215,
(the half of 42.43,) which set off on each side of C, and
from I to E, and from K to E, you will draw straight lines,
representing two meridians, which must be divided into the
same number of equal parts as CD contains; and through
the points of division, draw parallel lines, which will form
the figure I K F E, being a projection for one degree of lon
gitude, including six degrees of latitude.
413. Since the degrees must be so drawn, that the two
diagonal lines in each must be equal to one another, they are
to be projected in the following manner :
414. Take the distance from E to K, or from F to I; and]

129
setting one foot of your compasses, first in E, and then in F,
describe arcs L and M. Then set one foot first in I, and
afterwards in K, arid with the same extent, yon will draw the
arcs N and O. Take now the distance between E and F,
and set it off in the arcs described from E to N, and from F
to O; then take the distance between I and K, and set it off
from I to L, and from K to M.
415. Finally, draw the lines between L and N, and M and
O; divide them into 6 degrees, and draw parallels, from those
points, to the corresponding ones in the meridians I E and
KF. The same method may be pursued in drawing all the
other meridians and parallels the map is to contain,
416. If 1 he map be very large, so that the compasses will
not extend to the farthest degree, or from F to I, you may
draw one or more diagonals at once, and afterwards proceed
with the rest. Thus, when the parallelograms PG EN, and
HQOF are described, LIGP and KMQH, may be
drawn of the same magnitude.
417. Number the degrees of latitude up both sides of the
map, and the "degrees of longitude along the top and bottom
margins. Then, make the proper divisions and subdivisions
of the country, and having the latitudes and longitudes of
the principal places, you may easily set them down in the
map ; for any town, &c. must be placed where the circles of
latitude and longitude intersect.
418. Thus, if our map contain that part of Europe which
lies between the 39th and 45th degrees of north lat. and
between the 7th and 16 th degrees of east long, then Florence
must be placed at A, where the circles 43° 46' 30" north lat.
and 11° 3' 30" east longitude, cut each other; and Naples
you would place at B, on the sea shore, at 40° 51' 15" north
lat. and 14° 17' 30" east long. In like manner, the mouth
of a river, as the Tiber, for instance, must be set down; but,
to describe the whole river, you should take the latitude
K

130
and longitude of every turning, and note- them down, as
well as the towns by which it flows, and the bridges byhvhich
it is passable. '"'*•¦*¦
419. In the projection now described, the diagonals* being
all equal, the number of meridians causes no defect hi the
representation, as equal spaces, on the globe are represented
by equal spaces on the map; consequently, places lying in
the remotest degrees of longitude, are as truly represented
as those near the middle, and their distances will agree-' with
a common measure; so that your compasses extended be
tween any two places, and applied to the scale, will give you
the distance without further trouble.

PROBLEM II. i o
. . • "  ;.]
To draw the meridians and parallels of latitude for a map
of Spain, by method IsL See Plate XVI. Fig. 2d.
420. This country is between the north latitudes 36° and
44°, ' and extends from 10" to 23° of long, reckoning from
Ferro, or between nearly 3° in east long, and its western ex
tremity is about 9° in longitude west from London. We
shall use the reckoning of longitude from Ferro.
421. Draw the line A B, for a meridian passing through;
the middle of the country; and on this line set off 8° from
B to A, taking them from some scale of equal parts, A being
the north, and B the south point. Then, through A and B,
draw the perpendiculars CD, E F, for the extreme parallels
of latitude ; and A B being divided into eight equal parts or
degrees, draw the other latitudes parallel' to the former.
422. For the meridians, you will divide any degree in
A B, in to "60 equal parts, or geographical miles, as directed in
the construction of a diagonal scale, in articles 434 and 448.
Then, because the lengths of the parallels decrease towards
the poles, you will perceive from the table (page 10) shew-

Map <f .pari? of Italy with Covjioa and Sardinia

Ilaze,2yi

1*9 z
Map of Spain/

AC
i

C

11 22 13 14 IS . 26 ,J± 28 Iff 20 21 22 23 D

1

'44-
1 1 1
42 ¦ ' 1
= iX9
|41
1
I 1 j |
*l
1
W
|V
m
!^
l %Jk\
1
"' \l
sf 1 ^#^^
H/
 -jMllpl* E  .iiinnl 
litf
i
S JZ
22 23 24. 25 it : B 28 Iff 20 ZL 22. 23 T
!
army.
'cm, del,.
<?mar
,rcuTp
J,
f31
ing this decrease, that the number of miles answering to the
latitude of B, is 48-f' nearly, and this distance you set off
from B seven times to E, and: six or seven times to F; so
shall EF be divided into degrees. Again, from the same
table you find that the number of miles answering to a de
gree in the latitude, of A, is 43£ nearly ; which quantity
you take from the diagonal scale, and set off from A seven
times to C, and six or seven times to D. Then, from the
points of division in the line CD, to the corresponding
points in the linfe EF, you will draw so many-straight lines
for the meridians. ¦ • • ' '"
423. Number the degrees of latitude up both sides of the
map, and the degrees of longitude on the top and bottom
margins'; also, in some Vacant place, make a scale of miles
or of degrees, to serve for finding the distances of places
upon the map. '
4@4i Finally, make the proper divisions and subdivisions
of the country, and having the latitudes and longitudes > of
the principal: places, it will be easy to set them down in the
map; for any city, town, cape, &c. must be placed where ->
the circles of its latitude and; longitude intersect. For in
stance, Gibraltar, whose latitude is 36° 11', and longitude
12* '27', ~! will be at,G; and Madrid, "whose?, latitude5 is
40° 10', and longitude 14° 44', will be at M. In the: same
manner the mouth of a river may be set down ; but to de
scribe iJsi whole course, the latitude and longitude of every
turningy and of the towns arid; bridges by which it passes,
must- alsb be marked. The same is1 ; necessary for woods,'
forests, lakes', mountains, castles, &c. and the boundaries
exterior, are described by- setting down the remarkable
places on the sea-coast, and drawing a continued line through'
them alii '¦[ ' ¦ "''•"' i "U: -\
'.. .. ... ;•} •¦:•,.-; -.-:.' ¦:¦; •' ,»1 hi- v^
425. As maps of particular places are only parts of a
map of the world, you may project the sphere on the plane
K.2

132
of a meridian with pencil lines ; and any portion of the
earth's surface, may be included in a parallelogram, and the
superfluous parts rubbed out; as for example :
PROBLEM III.
To project a map of Asia by method 2nd. Fig. 1,
Plate XVII.
426. Project a planisphere by Problems I. and II., III. and
IV., or V. and VI., Section IV. and that your Asiatic map
may contain the Eastern Archipelago, and Australasia, draw
a line on the southern side of the equator parallel to it, .and
about fifty-five and a half degrees from it : raise on this line
two perpendiculars, about sixty degrees each from the central
meridian ; and let another line, parallel to the equator, pass
through the eightieth degree of north latitude; so shall a
parallelogram be formed, that will contain as much of this
planisphere, as is necessary for the proposed map.
PROBLEM IV.
To project a map of Africa by method 2nd. Fig. 2nd,
Plate XVII.
427. Project a planisphere as directed in the last problem,
and as Africa extends itself on both sides of the equator,
two lines may be drawn, parallel to that one, at about forty
degrees from it ; and two other lines, drawn parallel to the
central meridian, and about fifty degrees each from it, will
complete the parallelogram, that shall contain as much of
the planisphere as is necessary to bound the quarter of the
world for which the map is intended.

133

PROBLEM V.
To project a skeleton map of Africa by method 4rA.
Fig. 3d, Plate XVII.
428. In this construction the meridians only are curved
lines, the parallels are straight lines, and you would, there
fore, by Problems I., III., or V., Section IV. draw those
meridians. The axis meridian you will divide into eighteen
equal parts, and as Africa extends on either side of the
equator, you would set off two lines parallel to that one,
drawing them through the fortieth degree of north and south
latitude ; the other parallels are also parallel to the equator,
and by drawing two lines at the proper distance on each side
of the central meridian, and parallel to it, the parallelogram
that shall contain the proper quantity of the hemisphere for a
map of Africa, will be completed.
PROBLEM VI.
To project a map of North America by the second method.
Fig. 4th, Plate XVII.
429. Project a planisphere by Problems I. and II., or III.
and IV., Section IV. construct a parallelogram that shall
contain as much of the whole projection as is requisite; rub
out all the pencil lines beyond the parallelogram, and insert
the countries, &c. of that portion of the New World. If
your projection is large, it will be unnecessary to construct
a whole planisphere, a semi-planisphere only may be pro
jected,

134

PROBLEM VII.
To project a map of South America by rnelhod first.
Fig. 5th, Plate XVII.
4301 Project a planisphere as directed in the last Problem,
and enclose in a parallelogram the proper quantity of the
whole projection; efface the superfluous pencil lines, and
insert the countries, cities, rivers, &c. agreeable to their
respective positions as to longitude and latitude.
PROBLEM VIII. "'
To project a map of South America by the Mi method.
Fig. 6th, Plate XVII.
431. Project the meridians of a planisphere, as directed
in Problem I. Section IV. and through the ambit of each
parallel, or the equal divisions of the axis meridian, draw
the right-line parallels; then construct a parallelogram, that
shall contain the country in question ; rub. out the exterior
lines, and insert the places, &c.
PROBLEM IX.
To project a map of Europe, in which the meridians and
parallels shall be curve lines, according to method 2nd. See
Plate XVIII. Fig. 2nd*.,
432. Draw a base line GH, in the middle of which erect
the perpendicular I P. Assume any distance for 10° of lati
tude; and as Europe extends from 36° to 72° north latitude,
let I be situated at the 30th ; from which set off six of the

* NoteT.

H^jettwfu ^par^thrMabs on tTie plane of aMendum.
AFRICA

\ \ \ •¦ \ \ \ ! / / / / / / / / -
'-. \ ^ \ • * \ \ \ \ I / I / ' ' ,' / ¦'

ISORTB AME RICA

SOUTH AMUEICA

MEUICA

0^^

\\\\\\X\ ! \\W\\\

/

///// / / / /

//

•-' / / /

ismart jvufe>.

135
assumed distances to P, which will be the north pole. Num
ber the distances 40, 50, 60, &c.
433. About P as a centre, describe arcs passing through
the points of division in the perpendicular IP, which will
be the parallels of latitude.
434. Divide the space assumed for 10° of latitude, into
60 parts, by a diagonal scale, thus : Draw A B (Fig. 3)
equal to the assumed 10° of latitude; from A and B draw
two indefinite t;perpendiculars AC, BD; on AC set off six
equal parts, and through the points C, 10, 20. &c. draw
lines parallel to AB; join BC, which will be the diagonal;
number the divisions 10, 20, 30, &c. and subdivide them
into 10 secondary equal parts each, and your diagonal scale
will be completed.
435. Now^ to draw the circles of longitude, Took in the
table of geographical miles, page 10, for the number of
miles answering to 30°, which being 51.96, say 52, take
from the diagonal scale at b, by extending your compasses
from b to the diagonal line : let this distance be set off both
ways on the arc 30 30, from the central line or perpen
dicular IP. Do the same for 40, 50, 60, &c by setting
off according to the table, the measure of the miles from the
diagonal scale answering to those degrees; and through the
corresponding divisions on all the arcs draw curve lines,
which will represent the meridians; and about 30|° both
ways from the central meridian, raise the perpendiculars
G W, HV, and through the point 80, in IP, draw WV
parallel to GH. You will perceive that the parallel of 30°,
goes not beyond the parallelogram, yet, in drawing the
meridians, it is necessary to project it nearly to the arc of
a quadrant, and the same for 40°, 50°, &c. Number the
degrees of latitude and'longitude, which will complete the
diagram. 436. By inspecting this diagram, it appears that the meri
dians will never deviate much from a straight line, and it ,

136
will not create any very material error, if you should make
them perfectly straight, as in the following Problem.
PROBLEM X.
To project a map of Europe, in which the parallels shall
be curve lines, and the meridians straight lines, agreeably to
the third method. See Fig. 1st, Plate XVIII.
437. Draw the base line K L, bisect it in M, and raise the
perpendicular MN. Assume a distance for 10° of latitude,
and as Europe extends from 36° to 72° north latitude, let
M be situated at 30°, from which set off 5 of the assumed
10 degrees of latitude towards N. Number the distances,
40, 50, 60, &c.
438. The next thing to be done is, to take from a diagonal
scale the distances answering to the number of miles in the
latitudes 30° and 70°, which may be had from the foregoing
scale; or you may, if you please, construct another diagonal
scale, thus :
439. Let K L (Fig. 5th) be equal to the assumed 10° of
latitude: divide K L into six equal parts, which will deci
mate the assumed 10 degrees of latitude ; at the points K and
L, raise the perpendiculars KG, LH; make KG equal to
K L, and draw G H parallel to K L, which will complete
the square GHLK. Then divide KG into ten secon
dary parts, (as is shewn in Fig. 4th,) and through these 10
secondary divisions, (in Fig. 5,) draw lines parallel toKL
or GH; and from the points K, 10, 20, &c. in KL, draw
oblique lines to the primary divisions in G H ; as for example,
K10, 1020,2030, &c. which will virtually subdivide the
secondary lines into a diagonal scale, similar and suitable
in all respects to the former scale, known by the title, diagonal
scale. Fig. 3.
440. In the table of geographical miles, page 10, you

137
will find the number of miles answering to 30°, to be 51 96,
say 52 ; take the half of this, which is 26, from the scale,
Fig. 5th, and set it off from the point M both wtys, towards
a and b; the distance ab will be 10° of longitude.
441. Again, look in the table, page 10, for the number of
miles answering to 70°, which is 20 52, say20|; lake the
half of this, which is 10£, from the scale, Fig. 5th, and set
off from the point 70°, both ways, at c and d; the distance
ct/will contain 10 degrees of longitude in the latitude of
70° ; (cd should be drawn parallel to the base line KL;) and
then draw the two meridians ac,bd, which, if produced,
will meet in N, beyond the pole.
442. To set off the remaining meridians on the circular
arc 30°, take the distance ab, or from scale 5th, take the
line 2 a in your compasses, and about a as a centre, describe
the arc ef, and about b as a centre, describe the arc gh with
the same distance. Also, with the radius cd, or the line
c m on the scale, about c and d as centres, describe the arcs
m and n.
443. Take-in Fig. 1st, the diagonal ad in your compasses,
and with one foot in c, cross with the other the arc ef in the
point K, and with the s-me distance, and one foot on das a
centre, cross with the other the arc gh in /. With the same
distance, and one foot in a, cross with the other the upper
arc inm; and with the same distance, and one foot on b,
cross with the other the other arc in n ; then draw the two
meridians km, In. With the same three distances ab,cd,
ami the diagonal ad, perform the same operation for the
other ten parallelograms.
444. Produce any two meridians that are equidistant from
the central line M N, till they meet in a point, as acN, bdN.
The point of intersection N, becomes, then, a common cen
tre for describing all the circular arcs of latitude. For with
the radius N M, you will describe the arc a Mb; and with the
radius N40, the arc representing the parallel of 40 degrees;

138
also, on the same centre, the parallels 50°, 60°, and" 70"., are
to be described. Then, at about 30° from the central line
MN, draw the lines KR, LT, perpendicular to KL, and
through the point 80°, draw RT parallel to K L, which will
complete the parallelogram KLTR. You will observe,
that all the lines without the parallelogram, and the arcs at
m, n, k and /, together with those letters that note them, are
supposed to be drawn only in pencil lines, to be rubbed out.
445. Number the degrees of latitude and longitude, and
your skeleton map will be ready for the towns, cities, coun
ties, &c. being placed on it.
446. It is proper here to observe, that the pole is not the
fixed centre on which the parallels are to be described. It
was so in the last Problem— in this one it is about 6° and ^>ths
beyond the pole; and in that of Asia it is 30° beyond the
pole, varying, you will perceive, with every alteration of lati
tude and longitude.

PROBLEM XI.
To project the meridians and parallels for a map of Asia
by method third. See Plate XIX. Fig. 1st.
447. Draw a base line AB, on which raise the perpen
dicular CD of any length, and from C- set off nine equal
parts to P, which will be the pole; and if you go beyond P,
with three more of these spaces of the latitudes, the point 30
at D, will be the centre of the parallel of 60 degrees; that is,
the centre of the parallel of 60, as will presently appear, is
found to be 30 degrees beyond the pole, or, at the same dis
tance north of the parallel of 60°, as the equator is south of
it ; and the centre for this parallel is the centre for all the
others ; and it is evident that in this map the two diagonals
of each little figure are equal to one another, so that all the
parts of the map are of their proper size or magnitude.

i n
II!

liii
i ill!

ill liiiiii; !i iiiiii

— -- —

139
448. But you may construct your map from a diagonal
scale, of which HK, Fig. 2nd, would be equal to C 10, or
10 20, on the line C P of Fig. 1st ; and on H G Fig. 2nd,
are set off six equal parts, through which there are drawn
perpendiculars to KI, (KI being drawn parallel to HG.)
Those six primary divisions in H G, are subdivided into 10
secondary divisions each; and IH, the diagonal of the pa
rallelogram II K I G, completes the construction of your dia
gonal scale.
449. Then, to draw your meridians, with the quantity of
10° of latitude in your compasses, having previously about
D as a; centre, described the circle aCb, set off 10 equal
parts on the equator, both ways from C, and having describ
ed the arc c 80° d, through 80° in CP, draw EF at right
angles to C P. You may now find from the table of geo
graphical miles, the number of miles contained in 80° of
latitude, which you will take off a little beyond the number
20 (80° being 20° beyond 60°) in your scale, and set it off
ten times from 80 in the circular parallel 80, in the direction
80 c; and the same number of times in the same circular
parallel, in the direction of 80 b. Then, to the correspond
ing divisions in the equatorial circle, draw straight lines for
your meridians. These meridians will all terminate in D,
which is; therefore, the common centre for the parallel circles
of latitude (447)*.
450. For the parallels of latitude, take D for a common
centre, about which, with the radii D 10, D20, D23|, &c.
describe the circular arcs of latitude. Through the points
40° and 140°, raise the perpendiculars AE, BF, which will
complete the skeleton map of Asia.

* NoteU.

140

PROBLEM XII.
¦ To project a map of England according to method first.
451. England is situated between 2° east, and 6° 20' west
from Greenwich, and between 50° and 56° north latitude.
See Plate XIX. Fig. 3d.
452. Draw an indefinite base line AB, take any point o,
and raise the perpendicular oo, which will be the prime
meridian. Assume any distance for a degree of latitude,
and set off on o o as many degrees as are wanted, which in this
case are six ; but as a little space beyond the limits of the
country is generally left, set off seven. Then, through these
points draw lines parallel to AB, which will be the parallels
of latitude.
453. Respecting the degrees of longitude, it must be ob
served, that, on the equator they would be of the same
length as they are on a meridian, but, as has been demon
strated (31), they gradually decrease.
454. Construct your diagonal scale by dividing the as
sumed degree of latitude into sixty equal parts for a scale.
This scale is Fig. 4.
455. Then, the number of miles answering to 49 degrees of
latitude, is 39.36, say 39%, which take from your scale at a,
and set off seven times from o to A, and three times from o
towards B. The top of the meridian oo, is 56° of latitude;
opposite to which, in the table (33), is 33.55, say 33%, which
take from the scale at b, and set it off seven times from o to
wards F, and thrice from o towards E. Draw the meridian
lines to the corresponding divisions at the top and bottom of
your map, and of these meridians, oo is the prime meridian,
being that of London.
456. Finally, through A and B, draw AF, BE, perpen
dicular to AB; the 56th degree of latitude, or the line FE

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141
parallel to A B being already drawn : you will now have a
skeleton map of England, which you can easily fill up, by
inserting the differents objects that are necessary to be known
and viewed in such a map.
PROBLEM XIII.
To project a map of Russia by method third.
457. There is no diagram for this map, but it may be
constructed easily enough from a description of the prin
ciples on which its projection depends.
458. Draw any straight line for a central meridian, as
C D, in the projection of the map of Asia, Fig. 1st,
Plate XIX. and as the common centre for all the parallels
is found to be ten degrees beyond the pole, or 100 from the
equator, the point 10, between P and D, would be this com
mon centre, about which to describe the parallels of 80, 70,
60, and 50 ; but as it is necessary to go as far beyond 50 as
the forty-fifth degree, you may do so.
459. The 45th degree of latitude, by a reference to the
table, page 10, contains 42'43 geographical miles. Then
divide the assumed 10 degrees in C D into sixty equal parts,
by a diagonal scale, and on the parallel of 45 degrees, set off
in 10 spaces both ways from the central meridian, the quan
tity 42-43 taken from your diagonal scale.
460. The meridians being straight lines, and their com
mon termination being the pole P, (and not the 100th degree,
or point 10 beyond P,) you will draw straight lines from the
equal divisions in the 45th parallel, to the pole P.
461. You will observe, that the central meridian is that
of 100 degrees east longitude ; the other meridians extend to
190° east longitude, and to 10" of west longitude from this
primary line, on which the projection proceeds. . ,

142
462. To finish the construction, draw a straight line
through the 50th- parallel, and another parallel to that,- both
being at right angles to the central meridian ; and also draw1
two perpendiculars to those horizontally parallel lines, suf
ficiently distant from the central meridian to contain the
whole empire ; so shall a parallelogram be constructed, con
taining that portion of the meridians and parallels requisite
for laying down the countries in Europe and Asia subject
of Russia.

PROBLEM XIV.
To project a map of Russia by method secsnd.. ¦. , ; - *
463. Project a serai-planisphere by the globular, projectipn
of the sphere on the plane of the meridian,', and .jbounct as
much of this figure by a parallelogram as shall contain the
whole empire., There is no diagram for this projection,' butt
it must be very obvious from the preceding ones..

problem xv;.:1'
To project' a map of any country. '
464. When you have drawn a perpendicular line, and'
setoff the assumed degrees of latitude on it, as 'mentioned
in the foregoing Problems, take 2£S degrees in your com
passes, set one foot in the perpendicular, On the dot answer-'
ing to the 70th degree of latitude, extend'thd 'otbet foot
of yout compasses beyond the pOle, onrhe continued1 per
pendicular^ arid this point is the common centre for 'all
the parallels of latitude, they being" concentric' circles: f This
point is 8 degrees beyond the pole. '' ' ,'" " ¦'¦' V* '

143
465. The meridians must be laid down on each parallel,
from the table in page 10th, and their common termination
will be the pole; because, being subjected to the laws of the
decrement, shown in page 9tb, they are all, except the cen
tral one, curvilinear lines.
466. The assemblage of so many different kinds of maps in
this and the foregoing Sectionsj'will enable the student to esti
mate the properties and defects of each ; but it is very pro
perly observed by La Croix, that the term defect only refers
to the common way of considering maps ; for if we view
them with Euler and Lagrange* , as a transformation, of co
ordinates, it is always mathematically possible to obtain on a
mapj.j'all the geographical relations which may be required.
In fact, the position of different points of the sphere being
determined by their latitude and longitude, as the different
points of the plane are by two co-ordinates, if we assume on
a map,j lines subjected to a mathematical law, in order to repre
sent these co-ordinates, we Wall establish between the points
of the map and those of the sphere, such a relation that we
may assign on, the map the equation' of the lines, which cor
respond with circles, or even with any curves traced on the
sphere, and compare the relative spaces with each other.
Reciprocally it may be asked, What Ought to be, the nature'
of the co-ordinates of the map; that is, of the lines which
represent the meridians and the, parallels, in order that, the
parts of the map may have such and such a, relation, with
those of the sphere? In the solution of this last question by
the most refined analysis, Euler and Lagrange have deter
mined, a p riori, the construction of different kinds of, maps,
according to the qualities which they ought to possess. But it
is unnecessary further to enlarge on this way pf viewing maps.

* Memoued' Euler, Acta Academ. Petropol. torn. i. p. 1. Memoir*
de Lagrange, Academ, de Berlin, an nee 1799. '

144

SECTION x:

OF, THE CONSTRUCTION AND USE OF THE ANALEMMA*.
A Definition of it.
467. i\NALEMMA, in geography, is a projection of
the sphere in the plane of the meridian, orthographically
made by a straight line and ellipses, the eye being supposed
at an infinite distance, and in the east or west point of the
horizon. 468. Analemma denotes, likewise, an\instrument of brass
or wood, upon which this kind of projection is drawn with
a horizon and cursor fitted to it, wherein the solstitial colure,
and all circles parallel to it, will be concentric circles ; all
circles oblique to the eye will be ellipses; and all circles
whose planes pass through the eye, will be right lines. The
use of this instrument is to show the common astronomical
problems ; which it will do, though not very exactly, unless
it be very large. With the assistance of good maps, the
Analemma will not only solve many curious problems, but
be almost equivalent to a terrestrial globe.
469. I shall, first, give the construction of this instrument;
and, secondly, its use.
First. THE CONSTRUCTION QF THE ANALEMMA.
1st. To describe the points of the mariner's compass, the
primitive circle S W N E being given. Fig. 1st, Plate XX.
* See Note V.

x 145
!( 469. Divide the quadrant SE, into eight equal parts, and
draw through the points 11$, 22£, 33f, &c. straight lines
parallel to S N, and these parallel lines will divide the radius
OE, into a line of unequal parts; and each point of inter
section in OE, as 1, 2, 3, &c. is, reckoning from O the
centre, the extremity of the radius for that circle, which will
describe the first, second, third, &c. point of the compass.
470. Thus, with the radius O 1, you describe one point
of the compass; with the radius O 2, two points of the com
pass; and so on for the others. But for the proof,
' 471. As there are 360° in every circle, and 32 points in
the mariner's compass, 360° -f- 32 — 11° 15', being one
point; and 11° 15' + 11° 15' = 22° 30', being two points
pf the compass: and thus, by the continual addition of
11° 15' to the last quantity, you find that 90° contain just
8 times 11° 15'.
472. Those concentric circles, at the distance of ll£* each
from the centre of the primitive, shew the point of the com
pass on which the sun rises and sets, and on what point
twilight begins and ends.
t 2dly. To draw the parallels of the sun's declination.
^473. Divide the circumference of the circle (Fig. 3d)
into 360 equal parts, and through those points in the cir
cumscribing circle, draw lines parallel to the diameter, till
yon have come to 23\" on both sides of that equatorial dia
meter. These lines will represent the degrees of the sun's
declination from the equator, whether north or south, amount
ing to 23\ nearly. . On these lines you would now mark the
months and days, which correspond to such and such decli
nations. You will find these lines tolerably accurately laid
down on the diagram. But even the size of this diagram
does" not admit of having, every, day in the year inserted;

146
yet, by making an allowance for the intermediate days in
proportion to the rest, the declination may be guessed at
with tolerable exactness.

3dly. To draw the hour-lines.
474. Divide the quadrant ES (Fig. 2) into six equal parte,
each of which shall contain 15°, or one hour of time.
Then, through the points 15, 30, 45, &c. in the quadrant
ES, draw the lines 15 V, 30 IV, 45 III, &c. parallel to the
axisSN, and the radius OE will be divided into a line of
unequal parts, which you will transfer to the line O W, to
give you the hours I, II, III, IV, V, the primiiive circle
being VI. Or you might at once draw on O W, the parallels
75, 60, 45, 30, 15, which divided OE in the points I,
II, &c.
475. Now, as 60 minutes make an hour of time, a fourth
part of the space between each of the hour-lines will repre
sent 15 minutes, This distance, in using the Analemma,
your eye will readily guess at, and it is as great exactness
as can be expected from any mechanical invention, or, in
deed, as is necessary to answer any common purpose. ,
476. You have now to describe through the points NIS,
N II S, &c. five ellipses, which, with the plane of pro
jection N E S W, will be the hour-circles, designed to show
the time of sun rising or sun setting, before or after six o'clock,
in north latitude. The properties of the ellipse have been
already investigated in Section III. under the article, Ortho
graphic Projection; and the method of drawing an ellipse
in this case, has been shown by Problem D, page 93, Sec
tion VII. ; but I shall suggest a mechanical or geometrical
method of describing these semi-ellipses or hour-lines. We
have seen how the division of OE has been obtained; now,
let the point a (Fig. 2) be taken for a centre, and u 15 as
a radius, and describe a circle. Then, if you divide a qua-

147
drant of this last circle info six equal parts, and to each
fcixth part, draw a line perpendicular to 15 a 15, its diameter,
you shall have On a 15 its proper quantities, as accurately as
you have those on OE; and, doing the same by the other
lines, b SO, c45, d 60, e 75, you will obtain the exact points
through which the ellipses cut the several lines OE, a 15,
b30, &c. and you may form them, if drawn with a free hand ;
but I must again refer you to Problem D, page 93, and tlie
Appendix, for drawing an ellipse.

Of the Index of the Analemma.
478. The index is made thus : Take a slip of Bath paper
or pasteboard, and draw three parallel lines about £ of aa
Inch separate ; then cut out the inverse parts at the proper
length, so as to meet the primitive circle, as seen in that
on the 3d diagram; You may rivet your index on the
instrument, by a thread or fine wire, with a flat tin head, to
prevent its giving way by Use.

Secondly, of the use of the analemma.
479. In order to make nse of this Analemma, it is only
necessary to consider, that, wheh the latitude of the place,
and the sun's declination, are both north, or both south, the
sun rises before six o'clock, between the east and the elevated
pole; that is, towards the north, if the latitude and declina
tion are both north ;^or towards the south, if the latitude
and declination are both south.
480. Let us now suppose it is required to find the time of
the sun's rising and setting, — the length of the days and
nights, — the time when twilight begins and ends, — and what
point of the horizon the sun rises and sets on, for the
l2

148
Lizard point in England,
Frankfort on the Mayn, in Germany, or
Abbeville in France,
bn the SOth of April.
481. The latitude of these places will be found by the
maps, to be nearly 50° north.
To find the lime of the Sun's rising and setting, with the
Length of the' Days and Nights.
481. Place the moveable index so that its point may touch
50° on the quadrant of north latitude in the figure; then
observe where its edge cuts the parallel line on which the
30th of April is written. From this reckon the hour-lines
towards the centre, and you will find that the parallel line
is cut by the index nearly, at the distance of one hour and
21 minutes. So the sun rises at 1 hour and 21 minutes
before six, or 39 minutes after 4 in the morning, and sets
21 minutes after 7 in the evening. The length of the day
is determined, by doubling the time of his setting, which,
in this instance, is 7h- 21m# -f- 7h'2Im^== 14h-42m-
482. Observe how far the intersection . of the edge Of the
index, with the parallel of April the SOth, is distant from
any of the concentric circles, which you will find to be a
little beyond that marked two points of the compass; and this
shews that, an the SOth of April, the sun rises two poipts,
and somewhat more, from the east towards the north, or a
little to the northward of east-north-east, and sets a little to
the northward of west-north-west.
To find the beginning and ending of the twilight,
483. Take from the graduated arc of the circle, 17|-de-
grees, with a pair of compasses ; move one foot of the com
passes extended to this distance, along the parallel of the
SOth of April, till the other just touches the edge of the

- ¦ ^«tr^r:.» r.-.n. tr- ¦ jtr ¦:¦»>£ of JUn,7ismg KjOtzng. wpole ^ i%e Tmath -of itieDqys kM/Tos and tfi&
~"~ -¦ ~~ -¦¦-r7r -¦*¦-•- -^ tie Sun, w&^zZ^-*—^ *. *^ fir every Lyme ofZatttude. a?,

Fig. 5

for every Degree/ of the- Sum" Worth,

^^s* and/ South* J) eclijzatwris.

and

FlateJX.
Payc 148.

Fig. 4 .
if ^ ..;< - i

Drawn, iy j£ Jajmeron/

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149
index, which must still point to 50. The place where the
other foot rests on the parallel of SOth of April, then denotes
the number of hours before six, at which the twilight begins.
This is somewhat more than 3 hours and a half, which
shows that the twilight then begins soon after two in the
morning; and likewise, that it begins to appear near five
points from the east towards the north.
484. The uses of the Analemma may be varied in a great
number of ways ; but the example now given, will be suf
ficient for the ingenious student.
485. The most ancient treatise on this instrument now ex
tant, was written by Ptolemy, and printed at Rome in 1562,
with a commentary by Commandine *. ' Since that time,
many authors, as, Aquilonius, Jacquet, Dechales, &c. have
written on the same subject.
486. Sutton's Analemma of 18 inches diameter, has tha
meridians and parallels drawn through every degree, and
many of the principal stars laid down.

* Claudii Ptolemaei Liber de Analemmate cum commefttariis a
Frederico Cqmmandine, Romae, M.D.LXII.

APPENDIX; containing
1st. Some Methods of Drawing Large Circles and Ellipses ;
2dly. Directions for Colouring Maps ; Sdly. A Catalogue
of some of the best Maps ; and'4thly. G en eral rules and
observations for judging of the accuracy of different BJaps >
when a comparison is to be made between them, and places
inserted from one into another.

TO DESCRIBE LARGE CIRCLES,
The Beam Compasses,
made of brass, steel, or wood, from 3 to 4 feet in length, may be used.
Or if the radii of your circles exceed that length, the instrument may
be made longer. There must be two brass boxes or cursors exactly-
fitted to the branch, upon each of which may be screwed on steel,
pencil, or drawing-pen points, according as you have use for them.
One of the cursors is made to slide along the branch, and may be
made fast to it by means of a screw at the top; the other cursor is
fixed very near one end of the branch, -where there is a nut so fastened
to it, that, by turning it about the screw at the end of the branch, the
said cursor may be moved backward or forward at pleasure, to describe
the circle. These compasses may be had at the shop of any mathema
tical instrument maker. By the Carpenter's Rule.
Another instrument, called a Ship-carpenter's Rule, will be found
useful in large projections. This instrument consists of a strong piece
or back of wood, on each end of which is fixed a socket, to receive an
elastic spline, that may be forced to any degree of curvature by means
of three screws, one in the middle of the back.pjece, and the two
others between this central screw and the ends of the instrument. It is
usually sold at half-a-guinea.

152

By the Horizontal Compasses.
The improved Bevel or Horizontal Compasses, will draw an arc of
a circle through any three points, without the centre being known.
But as any instruments the student may use in projections, are described
in those works which treat of mathematical instruments and their uses,
J shall only detain him here with a description of the Cyclograph,
an instrument that will describe the arc of a circle to any chord and
versed sine ; but chiefly used in flat segments, or those whose curvatures
approach t6 straight lines.
There are several constructions of this instrument, but Peter IfipJioI-
son's improved. brass Cyclograph, is superior to any I have yet seen. It
may be thus described : — The two legs are connected by a folding
joint, and these legs in the instrument before me, are twelve inches
long, T%ths of an inch thick, and Toths of an inch broad. This part
of the instrument then is,, in shape and movement, exactly like a car
penter's foot rule of 12 inches. On these legs, at 5iin. from the joint,
there are ri vetted two slips, about 6|in. long, -j^th of an inch thick,
and i an inch broad ; these slips have' a folding joint, that connects
their other ends, so that when the legs are opened, this joint also
opens. That which serves as a rivet fpr each joint, is a piece of brass
projecting in the form of a pillar, above the upper face of the instru
ment about | an inch, and having a cylindric tube, through which
goes a rod, about T^ths of an inch in diameter. This rod is fixed
immovably, by a screw passing through the tube at the joint of the
major legs, but it slides easily through the tube over the joint of the
minor legs; and at this tube there is a nut, to fix the instrument -when
opened for use. In the end of this rod, next the head of the instru
ment, is a square pipe, that admits the shaft of the drawing-pen ox
pencil-foot, which is also made fast by a nut; and to keep the draw
ing-pen in the same plane with the major legs, is a guide that is fast
ened by a screw to one of the major legs. There accompany this
instrument, two triangular pieces of brass, having1 a steel point in each
corner, for- the purpose of fixing them, one in each extreme part of
the curve, in order to direct the instrument, or rather its head and
pen, in the curvilinear motion it shall take. It is now very obvious,
that there are three points, through which the proposed circle is to
pats : — one at the one brass guide, in the pane on which we describe
the circle ; another at the point of the pen or pencil; and a third at
the other brass guide. If, therefore, the instrument be moved laterally

153
close to the plane directors, the curve will be described accurately and
beautifully. See article 165 and 167.
The principle of this instrument is founded upon the 21st Proposition
of the Third Book of Euclid's Elements, in which it is announced and
proved, that, " The angles in the same segment qf a circle, are equal
to one another."
Nicholson's ". Centro-linead," is another instrument that will be found
useful in drawing lines or radii to a, point, when it is not known where
the centre is situated, because of being beyond your paper. When
the meridians are straight lines, as for example, in the map of England,
the instrument, which is nothing more than an improved kind of
bevel, being set to the angle, which the first meridian from that which
is perpendicular, makes. with the base line, and moved to the points
of division along the base line of your projection, all the meridians
may be drawn with ease and accuracy.

METHODS OF DRAWING AN ELLIPSE.
In any ellipses we have occasion to draw in orthographic or globular
constructions of maps, as, for example, in Fig. 5th, Plate XII. and
the Analemma, Fig. 3rd, Elate XX. the major axis is the longest,
and the minor axis is the shortest, diameter. Thus, referring to
Problem D, page 93, Pp is the major axis, and the minor axis lies on
a portion of the line EQ of Fig. 5th, Plate Xfl. ; and the ordinate
of any ellipse being a line drawn perpendicular to the major axis, II,
Fig. 3rd, Plate XX. will be a double ordinate qf the ellipse N I S.
You will perceive, that of all the double ordinates in this ellipse, the
longest is that which is next the minor axis of the ellipse ; and that of
the others, that which is near to this one, is always longer than one
more remote — they diminish gradually in departing from the minor
axis, till they arrive at the vertex of the ellipse, where they entirely
vanish. In any ellipses you may have occasion to draw in orthographic or
globular projections, the major and minor axes are always given.
Now, if you were to draw the ellipse by means of a string or thread,
from one extremity of your minor axis, as, for example, from If,
Fig. 2nd, Plate XX. with a radius equal to one-half of your major
axis, that is, equal, to OS or ON, describe an arc intersecting the
major axis in two points, which will be the foci of your ellipse: these
points fall about d and I. Next draw two pencil lines from the foci
of the ellipse to II, one extremity of the minor axis ; and fix a. pin in

154
each focus. Then, lo these two pins, you shall fasten the two end*
fef a string or thread, which must be exactly equal in length to the
'two pencil lines drawn from d and I to II; that is to say, when the
thread is stretched as much as possible from the foci or fixed points,
the middle of it shall just touch the end (II) of the minor axis^
Finally, take your pencil, apply it to the inside of the thread, and
leepmg the thread always stretched as much as possible, move your
pencil quite round from It to N and S, oh both sides of N S, so shall
the ellipse N 1 1 S be drawn, a portion of which is seen in Fig* 3, and
the complete ellipse is offered to your view by fdp in Fig.. 5th,
Plate XII.
Upon either the major or minor Axis, the following are
other Methods of describing an Ellipse, whose major and
minor Axes are mven.

¦minor Axes are given.

As by this method we can describe our ellipse either upon the major
orntinor axis, we shall, therefore, describe it, first, upon the major
axis. See Fig. 4th, Plate XX. Divide the major axis N S, into
any number of equal parts, say ten, and number them both ways from
»r 1, 2, 3,4, 5; through II draw the lineg/r parallel to NS, and
through N and S draw lines parallel to O Wj which will form the
parallelogram NS/g-. Divide S/, and Ng, into the same number
srf<equal parts, into which either half of the major axis is divided,
jtamely, into five; number these parts with numerals, 1,2,3,4,5.
From every point of the lines N it, S/, draw oblique lines to II, the
extremity of the minor axis ; and from H, the other- extremity of this
axis, draw through the points 1, 2, 3, 4, 5, of the major axis, oblique
lines, cutting the former in the points h, i, k, I, m, n, o,p, q, and the
points of intersection thus found, will be true points in the circum
ference of the required ellipse NHS; and you have now only to
connect all the points of intersection which you have just found, with
the extremity of the minor axis, and with the two extremities- of the
major axis, by a curved line. Let this curved line he done in
hsdian ink, rub out all the oblique and parallel pencil lines, and the
ellipse will present itself to your view complete, as in the semi-circle
NES. All the other ellipses in the projection, may be drawn in a
similar manner; but you will observe, that the greater the number
•f equal points of subdivision in the major axis, and the correspondent
number in the lines at Ng and Sf, parallel to WO, the greate*

155
will be the accuracy of the operation -.—accuracy which must entirely
depend upon the multiplicity of equal parts assumed in the lines
NS, Ng, and N/; and as the number of the points of intersection
will be in proportion to the number of equal parts in the major axis,
whenever the distance between them is very small, the curve may
be traced with the greater -degree of exactness.
But, secondly, by th'ts.method we can also describe our ellipse upon
the minor axis, and Fig. 5th, Plate XX. offers this scheme. The line
II II is the minor axis, and NS the major axis. The minor axis is
divided into ten equal parts, numhered both ways from 0, 1, 2, 3, 4, 5;
11/ and Hg are drawn parallel to NS;/g is drawn parallel to II II;
and divided into five equal parts each, numbered 1, 2, 3, 4, 5; that
js to say, into, the same number of equal parts into which O II is divided.
There are then drawn from S oblique lines, passing through the points
of equal division in the minor axis; and to N from each point of divi
sion in 11/ II g, there is drawn an oblique line, intersecting the former
jn the points h, i, k, I, vi, n, o, p, and q. These pqirits of intersection
do, therefore, determine the motion of the curve line to he traced from N
through IT, till it is terminated in the point S. The lower part of the
diagram exhibits both the half-ellipses, as will also the upper half
show the two others when your pencil lines shall be rubbed out. It is
possible to draw all the oblique lines at once, for determining the
points of motion for the curve; but as too great a number of oblique
Knes would render the figure very confused, it is always better to finish
Hie ellipse in detail.
• The student may probably feel as much predilection to this method
as myself, when informed that Major Pasley has adopted it in the
course of instruction originally composed for the use of the Royal
Engineer Department *. This method is the invention of Mr. Peter
Nicholson, who has given it in his Practical Geometry \.
It is hardly necessary to mention that mechanical instrument, the
Trammel, used by artificers to describe ellipses. This instrument is
shown by Fig. 6tb, Plate XIX. where 1,2,3, is the trammel rod; at
> is a nut, with a hole to hold a pencil; at 2 and 3 are two sliding
puts. Now, to use the instrument, make the distance of 2 from 1 half

* The Major's book is compiled and adapted to the mode of " British
instruction." ' + Nicholson's method applies equally to a circle, the hyperbola, and the
parabola ; for which, see his Work*.

156
th£ shortest diameter of your ellipses, and from the nut I to 3 equal
to half the longest, the points 2 and 3 being pat into the grooves of
the same size, move your pencil round at 1, and it will describe the
true curve of an ellipse. See also Note L.
Mr.Farey, a very ingenious mathematician in London, has lately
invented an instrument, by which to describe ellipses of any curvature.
This instrument may he had at Harris's the optician.

TO MAKE COLOURS FOR WASHING MAPS.
Colours proper for washing maps, are obtained by boiling different
kinds of wood, or staining substances, in water ; as, logwood for purple;
cochineal for red ; Brazil, madder, turnsol, &c, and the following
colours may be readily made; For Yellow.
Tate two ounces qf Avignon berries, (called also French berries,) a
dranvof alum, and one pint of rain water. Put them into an earthen
pot, and allow them to simmer over the" fire about, the space of half
an hour; then filter the liquid. This liquid, when cool, will be fit for
use; but should it be too strong, add water to it; or, if too weak,
simmer it again, for a very short time. This colour ought to be washed,
pretty pale, as it becomes deeper in drying.
For Blue.
Take an ounce of verdigrise, and three-fourths of an ounce of red
argil, three-fourths of a pint of clear rain water; put them together,
and let them soak for two days ; then boil them till they are dissolved,
and allow the liquor to settle before you pq.ur it off fqr use.
For Green.
A very excellent green may be made by mixing the yellow and
blue colours together; and verdigrise boiled in vinegar makes a very
good green ; but, as it dries deep, the wash must of course be very pals.

157

For Red.
An ounce of Brazil shavings infused in a pint of water, and boiled*
should have added to it a quarter of an ounce of powdered roach
alum; and, when cold, you may strain it off for use. Or you may,
in place of this tincture, use good lake, which will do full as well.
You will find that a little of this colour added to the tincture of Aviguoa
berries, will produce a colour similar to yellow ochre.
For Brown.
Boil bistre in urine. A good brown may be procured from gall- ,
stone as it is taken from cattle, and the real stone may be obtained in
anyplace where they are slaughtered. Ileart-of-oak chips boiled in
water, will produce an excellent colour, between brown and yellow-
ochre. Mr. Newman, artist's colourman, 24, Soho Square, has in liquids
the following colours: gamboge, verdigrise, carmine, purple, bistre
or vandyke-brown, Prussian blue, light green, dark green ; and of his
cake colours, red take, Prussian blue, Indian yellow, burnt terra-
siennae, and yellow ochre, may be used ; but it is hardly possible to
obtain a green from cake colours for maps.
In using cake colours, you must dilute them with soft water, observ
ing to mix them intimately together, till the desired tint of colour has.
been obtained ; then, with a brush' in proportion to the size of the
space you are going to colour, apply the wash steadily and quickly;
but do not suffer the proper limits to be passed, nor a greater quantity
of colour to be used than will evenly cover tlie space; as, when tco
much is used, it is liable to settle in particular places, and, by making
deeper tints in one place than another, the work will appear clouded.
You should also observe not to allow any two limits of the same
colour to touch each other, and endeavour to distribute your colours
so as to produce a pleasing effect on the whole ; but that taste which
quickly perceives excellencies or defects, so as to be soon delighted
with the former and disgusted with the latter, must assist you in
distinguishing and discriminating the most striking methods of giving
this finish to your maps (108).
The circular space beyond the outward meridian, as, jfor example,
that which contains the degrees of longitude in the Annexed
Figure, Plate XIII. may be coloured with burnt terra-sicnnE ;

158
and the space containing the rhumbs (286) may be washed with any
other colour.
The water should be coloured last, with a very light wsish of ver
digrise, and if it is not very expeditiously laid oh, it will be clouded
and offensive to the eye. The map should be damped with a clean,
moist cloth, laid Over it for a few minutes previously to colouring the
water. And should you have occasion to colour a printed map, the
colours may be kept from sinking, by wetting the back of the print
with a solution of four ounces of roach alum in a pint of spring
water, allowing the paper to dry from the water, before the colours
are laid on. This will not only prevent the colours from sinking,
but give them an additional beauty and lustre, and preserve them
from fading; and if the paper is not good, toash it three or four
times, suffering it to dry between every wash. The printing or let
tering the names of places, should be done when the colours on your
Inap are perfectly dry. To line Mttpg.
Cut the edge of your map even, so that if it consist of several
parts, it may join properly ; then take a piece of Irish cloth, the size
of the whole map, paste it with thin paste, and whilst wet, stretch
it by nailing the edges on a flat board; paste the several parts of
the map with thin paste, and lay them properly on the cloth ; attend
to it whilst it is drying, that it may not pucker, and when perfectly
^ry, cut the edges, and nail it on a roller.

A CATALOGUE OF THE BEST MAPS.
As a collection of good and accurate maps is of the greatest import
ance in the study of geography and history, I shall here subjoin a list
»f some of the best modern maps that have been published.
Those maps which may be collected for the purpose of forming an
Atlas, may be arranged under three heads, according to their size, or
the extent of their scale.
1st. Those which consist of more than six sheets, such as De Bovge's

159
map of Europe in fifty half sheets, and Cassini's map of France in one
hundred and eighty-three sheets.
idly. Those from four to six sheets, to which class belong several
maps of kingdoms.
3dly. Those from one sheet to four, wliich is the smallest size that
can answer the purpose of an atlas.

PLANISPHERES, OR MAPS OF THE WORLD.
Arrovisndth's, in four sheets, 1794. Smaller planispheres by Faden,
Harrison, &c. Northern and southern hemispheres by Faden, mat
sheet each, 1302. MERCATOR'S PROJECTION.
The best on this projection is that by Arrozvsmilh, eight sheets.'
That of Faden, in one sheet,

MAPS OF EUROPE.
Iff size. That of De Bouge, published at Vienna, 1799; or that by
Sortzmann, in" sixteen sheets, which is the better of the two. 2d sine.
By D'Anville, six sheets, 1754; or Arrowsmith's in four sheets. 3d sbe.
That by Faden, in one sheet.

Maps of England.
I. The Trigonometrical Surveys of the several Counties, published
by Lindley and Gardner. Some of the best surveys are published by
Faden. II. Cary's Atlas of the Counties, and his England and Wales
in eighty-one sheets. Smith's Atlas. III. Faden' a map in one sheet.
When the grand Trigonometrical Survey of England shall be finished,
as it is reported to excel in accuracy, abundance of positions, clearness,
and beauty, we shall enjoy one uniform map, like Cassini's map of
France.
Maps of Wales.
I. That of Evans, in nine sheets. I II. The maps in Pennant's Tours,
Evans's Cambrian Itinerary, and Aikin's Journey.

160

Maps of Scotland.
I. The Surveys of the several Counties. II. Aiiislie's nine-sheet
Map. Ill, Ah excellent map by General Roy, very scarce ; Pennarit's,
and Aiiislie's reduced Map, all in one sheet*
Maps bf Ireland*
/.I. Surveys of some of the Counties. III. A valuable map by Dr.
Beaufort, in two sheets, 1792 and 1797; the same reduced, one sheet,
Faden; and Taylor's^mt sheet, Faden, 1793.
Maps of Russia in Europe.
I. Maps of the different grovernments in the Russ character. The
same recent, in nine sheets. III. The Russian Empire, PetersbUrgh,
1789. Reduced map, one sheet, London.
Maps of Sweden.
I. Atlas of the Swedish Provinces, by Baron Hermelin, excellent,
and adorned with interesting prospects in Lapland. III. DelaRochette'sj
by Faden, one sheet. Maps of Denmark.
I. Maps of the Provinces, under the direction of Bygge. III. Faden's
maps of Denmark, Norway, and Sweden, in one sheet.
Maps of Prussia.
I. Poland and Prussia, by Zannoni, twenty-five sheets. Sortzmann's
Atlas, in twenty-one sheets. III. Sortzmann's reduced to two sheets
in France, and by himself to one sheet, in 1800.
Maps of Germany.
II. Chauchard's Map of Germany. I. Atlas of the Tyrol, twenty-
one sheets. Atlas of Bohemia, by Mutter, twenty-five sheets. Military
Atlas, twenty sheets. III. A Map of the Austrian Dominions, by
Baron Lichtenstern. Moravia, by Fenuto, two sheets.

161

Maps of the Netherlands.
i .
I. Ferrari's, in twenty-five sheets. II. Atlas de Department Belgique.
III. Ferrari's reduced Map, by Faden.
Maps of Holland.
II. Kep's Maps of the United Provinces. III. Faden's Map of the
Seven United Provinces. Maps of France.
I. CassinPs, mentioned above, and the Atlas Nationale in eighty-five
sheets. III. Faden's one sheet Map ; and a Map in Departments, by
Bellycime, in four sheets. France Physique, by Buache, one sheet,
which shews the mountains and rivers.
Maps of Spain.
Lopez's Atlas ; not, however, very accurate. II. A Map of Spain,
by Montelle and Chanlaire, jn nine sheets. III. Faden's Map in one
sheet. Maps of Portugal.
II. Geqffry's, improved by Rainsford, in six sheets} 1790. III. De
la Rochette's Chorographical Map, in one sheet, published by Faden,
1797. Maps of Italy and Venice.
I. Maps of the various states divided into provinces. States of the
King of Sardinia, by Borgognio, twenty-five sheets. III. D'Anville's
Map of Italy, improved by De la Rochette, in four sheets, published by
Faden. '
Venice, by SaiUini; Venetian Territory, Nplin, DeWitt, and Jaillot?
whose is the best.

162

Maps of Turkey in Europe.
II. Moldavia, by TSicarr, six sheets. Danube, by Mantfeld, seve*
small sheets. III. Bulgaria by Schenk. Bessarabia, by Gussefeld.
Greece, by D'Anville, and the Atlas of the Travels of Anacharsis.
Arrovismith's Turkey, two sheets. De la Rochette's Greece, one sheet.
Maps of Swhterland.
1. WeiSs's Atlas, published at Strasburgh in 1800, excellent. Weiss's
Seduced, one sheet.

MAPS OF ASIA.
The best general map of Asia, is that by Arrowsmitk, in four sheets,
published in 1801 ; and D'Anville's, in six sheets* may still be consulted
with advantage.
There are a few good maps of the individual countries ; but the fol
lowing are esteemed among the best.
Of China.
D'Anville'i Atlas, which Pinkerton thinks ought to accompany the
¦work of Du Hatde. A Map of China, by Arrowsmith. See also the
maps in Grosier's Account of China; but particularly those in the
Histairt Generate de la Chine. Of Tartary.
A map by Witsen, 1687, in six sheets, and one by DeWitt in one
sheet. Strahlenberg's is curfew.
Of Japan*
ItofarCs Map in one sheet.
Of the Birman Empire,
Tlie maps published in Mr. Symes's Embassay. The Geography of
lExterior India is very imperfect, but expected to be improved by the

163
Researches of Mr. Ddlrympk. For an outline of the coasts, the Cliarts
of D'Aspres are deservedly esteemed; and for Siam, D'Anville's Map
of Asia may be consulted. Of Hindoslan.
RennelVs Map, in four sheets. De la Rochette's, one sheet, good, third
edition, 1800. RenneWs Atlas of Bengal, and his Map of the Southern
Provinces, 1800. Faden's Peninsula of India, 179S, two sae.ets.
Of Persia.
There is no good modern map of this interesting country; but
La Rochette published a very beautiful one to illustrate the expedition
vi Alexander the Great. That of De Lisle, in one sheet, may be com
pared with the Asia of D'Anville or Arrowsmilh. Georgia and Armenia,
four sheets, 1788. Of Arabia.
There are some good partial maps in Niebuhr's Journey.
Of the Asiatic Islands,
There is an excellent Chart by Arrowsmith, in four sheets.
Of Australasia, or New Holland.
The best drawing is contained in ArroKsmith's Chart of the Pacific
Ocean, nine sheets. The same reduced, one sheet.
MAPS OF AFRICA.
The best general map of Africa is still that of D'Awdlh, three sheets,
1749, though some little addition may be made to it, derived from
the Journeys of Park and Brown. Wilkinson's four-sheet Map, pub
lished, in 1800. Major RenneWs partial Maps, may be consulted with
advantage. Of Abyssinia.
'There-is a good map in Bruce's Travels, that may be compared With.
«,hose of Tellcz and Ludolf, and the Africa of D'Anville.
U2

164
Of Egypt.
The best maps are those of the Delta, by Niebuhr, and Lower Egypt,
by La Rochette, 1802, one sheet.
Of the Makomedan Slates. ,
The best maps are those by Shaw, and a Chart of the Mediterranean,
in four sheets, by Faden. Of the Cape of Good Hopet
The best is Barrow's Survey.
Of the Eastern Coast,
There is a small map by D'Anville, called Ethiopie Occidentdle.
Of Madagascar,
Fkcourt has published a map, and Bellin has given a large one.
Of the African Islands,
There is a general map; and there are detached maps of the Isles
of Bourbon and Mauritius. MAPS OF AMERICA.
There is no modern general map of America that can be relied on.
The best is that of D'Anville, in five sheets, published in 1746 and 1748.
Mr. Arrotosmith has published an excellent map of North America,
on a very-large scale, but has omitted the Spanish dominions..
Of the United States,
Arrowsmith's, in four sheets, published in 1S02, is the best; and
there are very good maps of the individual provinces, in Morse's Ame
rican Geography,

165

Of the British Possessions in North America.
Besides Arrowsmith's mentioned above, Smith's Upper Canada, in.
one sheet, published in 1800, is very decent.
Of the West India Islands,
The best map is that of Jeffery's, in sixteen sheets, from which a
smaller one in one sheet has been reduced.

Of South America,
The best map is that published by Faden, in 1799, six sheets, from
3n engraving done at Madrid some years before *.
On Hydrography,
I must refer the reader to Pinkerton's Geography, where the best
account of charts is to be met with. See the 3d London octavo edition,
p.p. 795—797.

GENERAL RULES AND OBSERVATIONS FOR JUDGING OF
THE sACCURACT OF DIFFERENT MAPS, WHEN A COM
PARISON IS TO BE MADE BETWEEN THEM, AND PLACES
INSERTED FROM ONE INTO ANOTHER.
In employing this rule, it will be found that we disregard entirely
the projection employed in constructing the maps; for, in examin
ing different maps of Eur6pe, for example, we should judge of their
accuracy, by noticing what discordance exists in the latitudes and

-•* Tii e above catalogue differs but little from that in the " Jincjclopadiif
pritannica," See the article Geography, fifth edition.

166
longitudes assigned to the same place ; or by examining minutely
the distances of well-known places, such as the capitals of kingdoms,
ihe courses of rivers, and'the direction of great chains of -mountains,
the boundaries of the respective kingdoms and states, and the configura
tion of the checquered sea-coast, &c. we shall soon discover in what
they agree, and in what they differ, so as to form a pretty accurate
judgment of their comparative worth, and to which a merited pre
ference decidedly bfelongs. The student should use this rule especially
in classical geography, and the construction of those maps that would
.represent the old world at the period when imperial Rome boasted her
universal sway. But it is unnecessary here to give from Herodotus a de
tailed account of the map constructed by Aristagoras, tyrant of Miletus,
when he proposed to Cleomenes, king of Sparta, to attack Darius, king
of Persia, at Susa ; for even Ptolemy, who introduced meridians and
parallels into those maps which he copied from Marianus Tyrius, &c
has placed the longitudes of the places on the shores of the Mediter
ranean and remote from Egypt, much too great; and the celebrated
Peutingerian tables, or rmaps, though they Gontain an itinerary of the
whole Roman Empire, exhibit all places, except seas, (far .the Medi
terranean is there represented only as a broad river,) woods, and
deserts, according to measured distances, without any mention of
latitude, longitude, and bearing*. In fact, the countries are so dis
torted towards the north and south, in the map of Peulinger, that
they cannot be recognized. From a copy of the splendid edition
given by Scheele in 1753, thirty-five degrees of longitude in this map
occupy twenty feet eight inches, whilst thirteen degrees of latitude are
comprised within the space of one foot. But this is a circumstance
uot so remarkable, when we refiect.that the routes of the Roman armies
extended almost entirely from east to west; whence, to measures in
these directions, would be paid special regard ; whilst those between
north and sotfth 'quarters, whidh were'but partially -penetrated, might
almost entirely be neglected.
But in most of the ancient maps and charts, the most common
defect is, .to assign -too -much space between .places situated east and
west ;of each other ; for itthe latitude 'of a place .being more easily
determined than fits longitude, maps constructed from the relations

* :8ee ithe fHistoiy cof the cAcmleniy of inscriptions, vol. 18th, and the
History of the Academy Df;Siti8iicBs;£ard761.

167
?nd observatipns of travellers, would possess greater . accuracy as to
the latitude of places, whilst the erro.rs \a their longitudes would be
the greater in proportion as those places were further removed from
th* principal meridian. Now, to form any tJjing like a precise judg
ment as to the difference of longitude expressed in degrees, we must
attend to the diminution of the decrees between the parallels, and
wliich are ipfersected by the oblique line joining these ,two points,
But even this correction, is small in comparison of the error that more
modern observations have detected in these ancient determinations.-^
an error arising not only from the curvature of the earth being neg
lected, but because the itinerary measures were estimated in a vague
and incorrect manner, by day's journeys by sea or land ; as, for example,
in the map of Aristagoras, where on one straight line, called the royal
road or highway, were comprehended all the stations or places of
encampment from Sardis, the beginning of the route, to Susa, a dis.-
tance of 13,500 stadia, or 1687J Roman miles, of 5000 feet each, mak
ing thereby the total number of encampments in this whole route,
one hundred and eleven: and when in other instances these journeys
are reckoned by the windings of roads, through mountainous traets,
pi by fheir vicinity to the sinuosity of the shore, the difficulty of
giving a general outline of the map, must, from such data, become the
more considerable. With the materials to be gathered from the obr
nervations and relations of historians and travellers, the student, by
the assistance qflfilkinson's elegant " Atlas Classica," and M. Gosselin's.
work about to be mentioned, may complete a set of maps peculiarly
adapted to illustrate— 1st. The events of sacred history — 2dly. The ex
tent of the jurisdiction of the eastern and western churches— 3dly. The
accounts and descriptions which are met with in the classic authors —
and, 4thly. The histories of subsequent events, down to what are usually-
denominated the middle ages.
In comparing small maps by means pf the distances between differ
ent points — distances which, perhaps, have been the foundation qf the
map — we assume two points, whose distance shall serve as a standard,
to which the rest may be compared. In comparing the position pf
a third place with those on two different maPs? two triangles shqpjd^
be constructed on the same base, and the difference of their summits
will shew the discordance' between them : the middle point between
them is, therefore, to be chosen as the mean between them. Three
comparisons of this kind will give a triangle, and several will give a
polygon; and the mean position will be found, by taking the centre of
gravity of the figure thus formed. And to find the centre of gravity qf

168
a triangle, you have only to draw lines from each ahgle to the poin|
bisectingthe opposite side. This rule will serve for three determina
tions ; and when you have thus determined the distance of one place
from two others, supposed given in position, its longitude and latitude
may be determined, and the place transferred on the new chart or
map, whatever may be its projection. But, as hinted, I must, for
further information, refer the reader to M. Gosselin's " Recherches
sur la Ge.ographie Systematique et Positive des Anciens." See also
Notes A, S, and T.

MOTES and ILLUSTRATIONS,

Note A. Page 8.
The method of estimating the distances of places by longitudes and
latitudes, is of considerable antiquity, and was employed hy Eratos
thenes, who first introduced a regular parallel of latitude, whieh began,
at the Straits of Gibraltar, passed eastward, through the island of
Rhodes, to the mountains of India, all the intermediate places through
which it passed being carefully noted. Soon after drawing this paral
lel through Rhodes, which was long considered with a degree of pre
ference, Eratosthenes undertook to trace a meridian, passing through
Rhodes and Alexandria, as far as Syene and Mertie. > Pythias of
Marseilles, according to Strabo, considering the island of Thule as the
most western part of the then known world, began to count the lon
gitude from thence, while Marianus of Tyre placed their first meridian
at the Fortunate Islands, or the Canaries; but they did not determine
¦which was the most westerly of those islands, and consequently which
ought to serve as a first meridian. The parallel and meridian of Hali-
carnassus, according to the idea of Herodotus, the father of history,
as far as can be collected from hie history, Major Rennell makes to
intersect at right angles over the town of Halicarnasus *, in Doris, a
(Jrecian colony on the we,st coast of Asia Minor. This parallel cuts
the 37th degree of N. Lat. and the meridian corresponds with that
of 45£ long, ab Fort. Ins. versus orient. See Wilkinson's Atlas Classica,
plate XI.
; Among the Arabians, Alfragan, Albategnus, Nassir Eddin, and
Ulug Beg, also reckoned from the Fortunate Islands ; but Abulfeda
began to reckon his longitude from a meridian ten degrees to the east
ward of Ptolemy, probably, because it passed through the western ex
tremity of Africa, where> according to him, were situated the Pillars of
Hercules ; or because it passed through Cadiz, which was at that time
rendered famous by the conquests of the Moors in Spain.

* This place gave birth to Herodotus, who flourished about four hundred
«nd forty-five years B. C.

170
When the Azores were discovered by the Portuguese in 1448, some
geographers made us>e of the island of Tercera as their first meridian.
Other geographers, as Blaen, father and son, placed the first meridian
at the Peak of Teneriffe, a mountain so far elevated above the sea,
that it may be easily known by navigators ; while others have made
the island of St. Philip, one of the Cape de Verde's, the first meridian,
because they conceived this to be the place wheie the magnetic needl?
ftsd no variation. For a long time it -wns customary to reckon the
longitude in most countries from theisle of Ferro, one of the Canary
Jates; but it is now customary for each nation to reckon the longitude
either from the metropolis pf the country, or from the national observr
story situated near it. Thus, in France, Paris is , the first metidjaiij
»»»d in Great {kjtain, the RoyaJ .Observatory at Greenwich.
As, in several good maps, the isle of Ferro is still used as a first
meridian, it may be proper to remark, that the observatory at Green
wich lies 17" 45' to the east of Ferro. Hence, it is very easy \q reduce
¦the longitude of Ferro to that of Greenwich : fpr, if the longitude
required be east, we have only to subtract 17" 45' from the longitude
«rf Ferro, aad ,the remainder is the longitude east from London: 0n.
the other hapd, if the place be west from Ferro, we obtain the longir
'Indie-- west from London, by adding to that of Ferro 17° 45/. If the
place lies between Ferro and London, its longitude from London will
fee obtained, by subtracting its longitude .east .from Ferrp from 17° 45',
It is evident, that by .the reverse of this iinetbod, we may reduce thlj
Jor>g.Uude from London to that of Ferro,
Note B. Page 17.
Having said so much on the nature of the terrestrial globe, we
shall subjoin the following practical rules for the choice t>f ^globes;
atrd, - .
I. Thepa.pers or gores should be well and neatly pasted on the
globes, which may be known by the lines and circles, exactly meeting^
and continuing all the way even and whole; the circles not breaking
into several arches, nor the papers either coming short, or lapping
over que another.
2. The colours should be transparent, and not laid too thick upon
the globe, to hide the names of places.
3. T-he -globe -should hang -evenly -between -the -biaaen meridian
and the wooden, horigon, not inclining either to one sid^or the other.
4« The globe should move as (Slose to the horizon and. the meriduil

in
as it conveniently may, -otherwise there will be too much trouble to
find against what part of the globe any degree of the meridian or hori
zon is.
5. The equinoxial line should be even with the horizon all round,
when ftie north or south pole is elevated 90» above the horizon.
6. The equinoxial line should cut the horizon in the east and
west points, in all the -elevations of the pole from zero or 0» to 90°.
7. The degree of the brazen meridian marked o, should be exactly
over the equinoxial line of the globe.
8. Exactly half of the brazen meridian should be above the hori
zon, which may be Jmown fay bringing any of tlie decimal divisions
on the meridian to the north point of the horizon, and finding their
complement to 90» on the. south point.
9. When the quadrant of altitude is placed as far from the equator
or brazen meridian, as the pole is elevated above, the horizon, the
-beginning of the degrees of the quadrant should reach just to the
iplane surface of the horizon.
10. When the index of the hour circle, provided the globe be fur
nished with one, passes from one hour to another, 15" .of the equator
•wiust pass under the graduated edge >of the brazen meridian ; and
when the globe circle is without an index, the division for one hour
and 15° on the equator should be exactly cut by the .graduated edge
of the brazen meridian.
11. The wooden horizon should be made substantial and strong,
it being generally observed, ithat in most globes the horizon is ,the first
•part that fails, on account of ihs having been made too slight.
The best globes are those by Adams, Cary, and Bardin. The last
maker covers his from drawings by Arrowsmith, with the newest dis
coveries ; the celestial globes are also executed with great care and pre
cision. In Cary's celestial globe (1808) the constellations are only
marked by boundary lines, and the eye is not distracted with the ridi
culous figures of animals, '&c. Some astronomers, however, prefer
the ancient figures, on account of speedyiand, accurate reference.
Note C. Page 36.
ilihe Rev. Dr. Vince uses this projection in the construction of
solar eclipses. , See his " Complete Astronomy," Vol. I. second edition,
quarto, 1814. Consult also Dr. Hutton's " Mathematical Dictionary,
on the Laws oT fhe Orthographic Projection," -Vol.H. Page-293, -Ed+f-ien
1795. Dr. Rees's Encyclppaedi3,<article„ "Praj,eet|onvOfthe,Sphere/' &<j.

172
I cannot conclude this note, without recommending to the student's
attention, Emerson's " Treatise on Projection ;" that in Robertson's
*' Navigation," Bishop Horsley's "Elementary Treatises," and Lacaille's
and Vince's" Astronomy." There is also a very elegant one in Puissant's
<£>Traite de Topographie, d'Arpentage, et de Nivellement ;" as, likewise,
in Bonnycastle's '- Trigonometry," Dr. Hutton's and Mr. Barlow's
ff Mathematical Dictionary," and Mr. Ktitlis ''Trigonometry."
Note D. Page 40.
Of all the ancient geographers, posterity is most indebted to Ptolemy,
who produced a work much more scientific than had ever before been
written on this science; a geography in eight books, which must ever
be considered as one of the principal monuments of the labours of its
author. In this work there appear, for the first time, an application
of geometiical principles to the construction of maps; the different
projections of the sphere; and a distribution of the several places on
the earth, according to their latitudes and longitudes. This work
must have been the result of a great many relations, both historical and
geographical, that had been collected by this indefatigable man. It
lias passed through numerous editions.
Ptolemy not only rendered to geography the greatest service, in
collecting all the known longitudes and latitudes of different places,
and laying the foundation of the method of projections for the con
struction of geographical charts ; the astronomical edifice he raised,
subsisted nearly fourteen centuries, and now that it is entirely destroy^
ed, his " Almagest," considered as a depository of ancient observa?
tions? is one of the most precious monuments of antiquity *.
Note E. Page 44.
¦ Mr. Bonmjcastk, in a note, page 59, '.' Mensuration,'' says, 'f this
construction is the invention of Rhenaldinus, and was first given in his
second book, ' De Resol. &c. comp. Mathem, p. 367.' The rule for
polygons in general, is onjy an approxiinatiqn, but holds true in the.
equilateral triangle and hexagon." But you will observe that Mr. Bon-?
nycastle does not divide the radius, but describes an equilateral triangle.
Stone, in his " Mathematical Dictionary," examines this trigonomei
_  ,  ,, .fu4
• La Place's System of the World, p, 281. V9L l|«

173
Ideally. See " Regular Polygons," in that work. The series is the!
only general method, however, than can be applied in the investiga
tion of this property. See Mr. P. Nicholson's " Architectural Dic
tionary," page 340. Indeed, this gentleman was the first who sug
gested to me the idea of taking |ths of the radius for the point X, and
he assured me that it strqek his mind, many years ago, as applicable
to the globular projection of the sphere, though he never reduced '
it to practice till the present (1814), in, the article " Projection of the
Sphere," for Dr. Rees's Encyclopaedia. This projection is not new,
but, on the contrary, very ancient, for Ptolemy mentions it in his
Geography, and so does Blundevill in his Exercises.
Note F. Page 46.
Of the regular polygons, three only are susceptible of perfect adapt
ation, and capable, therefore, of covering, by their repeated addition,
a plane surface. These are the equilateral triangle, the square, and
the hexagon. The angles of an equilateral triangle, are each twp-thirds
of a right angle, those of a square are right angles, and the angles of
a hexagon are each equal to four third-parts of a right angle. Hence,
there may be constituted about a point, six equilateral triangles, four
squares, and three hexagons; But no other regular polygon can admit
of a like disposition ; and, on this principle, the adaptation of patch
work, and the construction of tessellated pavement, are founded *.
Note G. Page 52.
By separating the principles of the projections of the sphere front
these geometrical and trigonometrical practical methods of projection,
the student is not brought to encounter one undistinguishable mass,
requiring superior perspicuity to disentangle perplexity and regulate
confusion. But even this part of our work involves the nicest prin
ciples of science — principles,, which being purely geometrical, con
dense the most rigid theory with the most rapid facility of practice,
and admit the mind, delighting in assiduity, to a richly diversified
banquet. '
With respect t.o the method by which I determine the centres of
the meridians, in Problems I. and VIII. I have to observe, that it is

* Prof. Leslie's Geom. p.p. 120 and 420, second edition, Edinb. 1SU,

174
the most expeditious of stny yet presented to the public : it is equaf
to Arroii'smilh's in Problems III. and IV. or the trigonometrical
methods, in Problems V. and X. for the formulas upon which it de
pends are strictly trigonometrical. This method presented itself to
my mind, from the labour I used to be- at, in dividing each of the lines
drawn obliquely from the point of view, to the equal divisions of
the equator, in the globular, and the semi-tangetical divisions of the
equator, in the stereographic projection. The moment it occurred,
its simplicity and facility were too powerfully felt to be resisted —
I have adopted it without hesitation, in preference to the tedious
method which is hinted at above, and which is usually given to effect
the same geometrically.
Note H. Page 59.
¦problems III. and IV. are really elegant, and do great credit t«
Mr. Arroxvsmith, who, with great labour and exactness, drew all the
meridians and parallels of latitude to every degree on two hemispheres,.
which laid the foundation of his excellent four-sheet map of the world.
His pamphlet, called " A Companion to a Map of the World," ex
plains the projection, and contains some valuable information. A
satisfactory account, and a just critique of the maps and charts of
this deservedly-esteemed geographer, will be found at the end of the
third octavo edition of Pinkerton's Geography, published in 1811.
See also the Encyclopaedia Britannica, fifth edition, Vol. IX. Part II.
article, Geography, No. 126> page 546, where, " of the latest maps and
charts," those constructed by Arrowsmith, are declared to be "in the
greatest estimation." Note I. Page 77.
It is not certain by whom this analogy was at first discovered, but
the discovery seems to have been made by accident, in some such
similar manner as the curious properties of the crescents or lunula?
were discovered by Hippocrates of Chios, in his attempts to square
the circle.
The analogy between the increased meridian and a scale of loga
rithmic tangents, was first published and introduced into the practice
of navigation, by Mr. Henry Bondx by whom this property is men-.
tioned in an edition of Norwood's " Epitome Qf Navigation," published
about 1645.

175

Note K. Page 77.
It was the great study of our predecessors to contrive such a chas*
in piano, with straight lines, on which all or any parts of the wortd
might be truly laid down aecoirding to their, longitudes and latitude*.
A method for this purpose was hinted at by Ptolemy, nearly two thou
sand years ago ; and a general map, in such an idea, was made hw
Gerardus Mercator, about 1550, Wherein the degrees of latitude jweiie
increased from the equator towards each pole; bat the principle*
were not demonstrated, nor was a ready way shown of describing
the chart, till Wright, an Englishman, about the year 1590, dise»vei:e€
the true principles, and explained how to enlarge the meridian line^
iiy the continual addition of secants, so that all the degrees of lon
gitude -might be proportional to those of latitude, as on the .globe
itself, which renders this chart in several respect-s far more convenient
for the navigator's use tban the globe itself, and which will show .tauhy
the course and distance from place to place hi all cases of sailing.
This skilful, but unsuspecting mathematician, communicated fr«e%r
the principles of this projection to one Jodocus Hondins, an engraver,
who, contrary to his honest faith and engagement, published the sante
as his own invention. This occasioned Mr. Wriglit, in the year 1599,
to exhibit "his method of construction, in his book, entitled, " Correc
tion of Errors in Navigation ;" in the preface to which may be sees
Ms charge and proof against Hondius; and also how far MerGOt or has
any light to share in the honour due for this great improvement ia
geography and navigation *. But Mercator's name will, by this map,
be transmitted to future ages with deathless fame.
This Unassuming, wonderful genius — Wright, conceived a glob*
or sphere, with the susual delineations, posited in a cylinder, to toe
-blown like a bladder, till every part of the surface touched Or adhered
to the tcyliader. Thus would each parallel have the same diameter
with the equator or cylinder, and the meridians becoming parallel
ito each other, would in all latitudes have the same distance as on the
equator. In short, the rhumbs and all other circles of the globe, would!
¦unbend themselves into straight lines.

f Se0 the Philoso. trans, for JT58, Vol. L. Part II. p., 565.

176

' Note L. Page S4.
The proportional compasses is an instrument made of two equal
fclips or pieces of brass or silver of any length, the breadth and thick
ness of which must be proportionable, with four equal steel points.
The slips .are, connected by a cylindric pin, which, when they are made
to coincide, is moveable along a slit through both pieces, so that the axis
pr centre of the pin, and the two extreme points of each piece, may
lie in the same straight line, and fixed at any point, which will keep
its situation in each of the pieces, whether the instrument be opened
at any angje or shut. The two surfaces of the instrument, which
appear wh^n it is shut, are the sides ; and the four parts of the instru
ment from the centre of motion, are called legs.- Hence, any pair
of legs will be equal to each other, and of the same invariable length,
whatever angle these legs make each other.
The proportional compasses have usually laid down upon their
faces the line of sines, the line of chords, &c. This instrument may
be employed — lit. To divide a given line into any given number of
equal parts from 2 to 10 or 100; idly. A right line being given, and
supposed to be divided into one hundred equal parts, to' take any
number of those parts; Zdly. The radius being given, to -find the
chord of any arc under 60°; itltly. The radius bejing given, to fiud
the sine of any number of degrees ; 5t}ihj. The radius being given,
to find the tangent of any number of degrees, not above 71". But
the proportional compasses may be applied to. perspective, some
branches of practical geometry, and different parts of geometrical
architecture. For these applications I must refer the^reader to the
article " Proportional Compasses," in Rees's Encyclopaedia, written v
by my friend Mr, Peter Nicholson.. ,..',.¦
In conjunction with the proportional compasses, a pair of screw
dividers, and light beam compasses about a yard long; several ellipses-
rulers may be used for, drawing lines, which are too large to be struck
with beam compasses; and these ellipses-rulers will be found par
ticularly useful, when large partial maps are being done, in curved
lines, from the table of proportion of latitude, and decrease of lon
gitude, pages 10th and llth. By the method I have given (page 93^
ef drawing an ellipse, or from the art-ide Ellipse- in the Appendix,
the student may make out of stiff cards, card boards, ,or^ thin holly-
wood, ellipses rulers and sweeps, which will answer his purpose as well
as an expensive brass ellipsograph.

177
There are various simple means for reducing or enlarging maps,
besides what I have mentioned; but there is, perhaps, none more
effectual, than that of proportionate squares. The method is this :
Divide the original map or picture, into any number of equal squares ;
then divide the paper upon which you are going to. work, into the
same number of equal squares, either greater or less than those in the
original, according as the pi^ce is to be. enlarged or contracted ; then
draw or project the several parts of the piece as they fall in the squares
of the original, in the corresponding squares of the paper you are.'
drawing on: then outline it with Indian ink, and rub out the black-
lead maiks of the squares; insert the minutiae, and lay on your colours,
so as readily, and at a,; cursory view, to render the ,map. still, more
intelligible; for whether your colouring be more or less beautiful,
it should always assist us in distinguishing the face of a country at a
glance. See Note T, p? 186.
Note M. Page 89.
The logarithmic spiral may be drawn from a series of points to be,
found in the curve,; the centre, and two opposite points in a straight
line passing through the centre, being given in the curve. The chief
instrument in performing this, is the proportional compasses. The
principle is this: Two right lines being drawn, cross each other at
right angles, and two others at 45fl, pass through the point of intersec
tion; a distance is then assumed for a radius, and a mean proportional
is found between the radius and the fifth point in the curve: this mean
gives the third point in the curve. Also, a, 'mean proportional is
found between the radius and third point in the; curve. A straight
line is then drawn equal to the radius, and the- cpmpass.es are set in
the ratio of the radius to the first proportional, or point in the spiral
curve. The radius is taken between the longer legs of ,ttie compasses,
and the distance hetween the shorter gives .the n>st proportional in the
curve; then, the longer legs" set,, to. this proportional, determine by
the space between the shorter ;legs,--the third proportional. Again,
this third proportional taken between the-lpngerjegs of the instrumept,
expands the shorter legs to the fourth proportional; and so on in
regular geometrical proportion; and, therefore, if a curve be drawn
through all these points of revolution; it will be the spiral required.

17S

Note N. Page 90.
It may be worth the student's pains and labour, to examine the
projections of the sphere, and the constructions of maps, in Dr. Rees's
Encyclopaedia. Mr. Peter Nicholson, who furnished these articles,
has reduced Brook Taylor's " Perspective," to projection generally,
and to the projection of the sphere particularly"; and the extreme
simplicity with which these projections are given, proves how very
successfully Nidholsdn'' has exerted his ingenuity, in converting par
ticular to general rules ; wliich, by their novelty and singular beauty,
must prove as highly interesting, as they will be deservedly valued.
The horizontal projections contained in this section, are stereo
graphic projections ; but it was observed in page 50, that the globular
is capable of giving distances, as well as the stereographic ; and
Boidlanger's Map of the World (1760), is on the horizon of a point,
45° of the height of the pole towards the north. In 1774, Father de Gy
published one similar, projected on the horizon of Paris. These
maps present, under one point of view, the four parts of the world,
which, as Fleurieu says, "nature has assembled under the same hemi
sphere." There are planispheres published at Vienna, stereographically
projected on the horizon of the place 6f publication.
Note O. Page 99-
Horizon, in geography and astronomy, is a great circle of the sphere,
dividing the world into two parts or hemispheres; the one upper and
visible, the other lower and hid. The word ' is pure Greek, and
literally signifies hounding or terminating the sight; whence it is called
finitor— finisher.
The horizon is either rational or sensible.
The rational, true, or Astronomical Horizon, which is also called
simply and absolutely the horizon, is a great circle, whose plane
passes through the centre of the earth, and whose poles are in the
zenith and nadir. It divides the sphere into two equal parts or hemi
spheres. The sensible, visible, or apparent Horizon, is a lesser circle of the
sphere, which divides the visible part of the sphere from the invisible.
Its poles are likewise the zenith and nadir ; and, consequently, the
sensible horizon is parallel to the rational, and it is cut at right angles,
and into two equal parts, by the vertical. These two horizons, though

179
distant from each other by the semi-diameter of the earth, will appear
to coincide, when continued to the sphere of the fixed stars, because
the earth, compared with this sphere, is but a point.
The sensible horizon is divided into eastem.and western.
The eastern or ortive Horizon, is that part of the horizon wherein
the heavenly bodies rise.
The western or occidental Horizon, is that wherein the stars set.
By the sensible horizon is also frequently meant a circle, which
determines the segment of the surface of the earth, over which the
eye can reach; called also the physical horizon. In this sense we say,
a spacious horizon, a narrow, scanty horizon.
It is manifest, from Fig. 2, Plate XIV. that the higher the spec
tator is raised above the earth, the further this visible horizon will
extend, as the respective distances A D, B A, will be greater.
On account of the refraction of the atmosphere, distant objects on the
horizon will appear higher than they really are, or appear less depressed
below the true horizon S S, and may be, seen at a greater distance, espe
cially upon the sea. Legendre, in his Memoir on Measurements of
the Earth, in Mem. Acad. Sci, for the year ] 787, says, that, from several
experiments, he is induced to allow for refraction one-fourteenth of the
distance of the place observed, expressed in degrees and minutes of a
great circle. Thus, if the distance be 14,000 toises, the refraction will
be 1000 toises, equal to the fifty-seventh part of a degree, or 1' 3".
Note P. Page 109.
When the sun's amplitude is to be found, the latitude and sun's
declination being given, say,
As the cosine of the latitude,
Is to radius ;
So is the sine of the sun's or star's declination,
To the sine of the amplitude.
Thus: were it required to find the amplitude of the sun in the latitude
of London (51° 32') : the declination being 23° 28',
Ascos.5l°32'  97938317
Is to -radius  10-0000000
So is sin. dec. 23° 28'  .,  9-6001181
To the amplitude 39° 48'  ,  9-806S864
And this is of the same name with the declination ; viz. north, when
the declination is north ; and south, when it is south.
n 2

180
The azimuth, at the time of the equinox, may be found trigono-
metrically thus:
As radius, <
Is to the tangent of latitude ;
So is the tangent of the altitude of the sun or star,
To the cosine of the azimuth from the south.
To find the azimuth of the sun or a star at any other time, consult
any of our elementary treatises on astronomy.
Note Q. Page 111.
The principle of developement is very fully treated in Nicholson's
" Architectural Dictionary ;" see the words " Carpentry" and " Dome."
Plate XIII. belonging to the former article, and Plate If. answering
to the latter, show at once the data upon which this principle is found
ed. What Langley, Swan, Pain, and Price, had said on the principle
of covering the inscribing and circumscribing spherical dome, the
same ingenuity which the previous publications of Nicholson had
made familiar to us, legitimately applies to the developement of the
terrestrial globe oh a plane surface, as that of a planisphere.
Murdoch's method is a mixed scheme of concentric circles and
straight lines; Nicholson's consists entirely of arcs of circles and cur
vilinear lines. Though I have adopted Nicholson's application, I
have, as will appear from the investigation, treated it purely geome
trically, agreeably to my own ideas of the principles on which it rests.
1 Note R. Page 124.
It is to the develbpements of the globe, that we must refer the con
struction of spindles or gores, which are drawn upon paper, in order
to cover globes of a moderate size. The surface of the globe is divided
into twelve or eighteen parts, according to the size of its diameter/ by
drawing meridians from 30a to 30°, or from 20° to 20°. The space
comprehended between two of these meridians, having a very small
curve in regard to breadth, may be considered as forming part of a
cylindric s-.-rface, circumscribed on the sphere, according to the meri
dian which divides it into two equal parts. This meridian being
developed in bearing perpendicularly- on each side, according to
the law of ordinates, the half-widths of the portions or parallels, com
prehended between the meridians which terminate the spindle, we
obtain the form of its entire developement For, suppose that O P,

181
Fig. 3d, Plate XV. of the axis of the curve, is equal to the fourth-
part of the circumference of the intended globe (397), the intervals of
the parallels on the axis O P, are all equal ; the radii of the circles
10, 20, 30, &c. which represent the parallels, are equal to the cotangents
of the latitude, and the arcs of each, such as 30, are nearly equal to the
number of degrees that correspond* to the breadth of the gore, mul
tiplied by the sine of the latitude: thus, there will be found no dif
ficulty in tracing them ; but the principal difficulty proceeds from
the change which those parts of the gores undergo, when they are
glued upon the globe; as, in order to adjust them to the space which
they ought to occupy, it is necessary to make the paper less on the
sides than in the middle, because the sides are too long. Bion, in his
" Usage des Globes," torn, iii., and Robert Faugondy, in the viith vol.
of his "Encyclopedia," and LaLande, " Astronomie," torn. iii. p. 616.
third edition, have gone fully into the methods employed by artists
for engraving these gores. In glancing at the figure referred to above,
you are to view that part of it included between the two meridians
M and N, on each side of the central meridian, as the semi-gore ; but
even this does not represent the exact construction. However, when
it is impossible to prescribe a rigid operation, sufficient to make a
plane which shall cover a curved surface, it will be here necessary
only to inform the reader, that the curved ordinates 10, 20, 30, &c.
of the semi-gore MPN, have for their respective radii, the points
b, c, d, &c. of No. 1 . Fig. 2d. And these successive curved ordi
nates, described from different centres, we should name eccentric
ordinates. But the gore i&. sometimes truncated at the two extre
mities, at 15° or 20" from the poles ; and these two zones are drawn
apart, as if they were flat. The student, to form some tolerably fair
notion of the making. of gores, has only to visit a cooper's workshop,
and observe the form and manner in which the staves of a cask are
prepared and put together. The principle of shaping the staves, is
analogous to that of constructing the gores for a globe.
-Note S. Page 125.
The examples of particular constructions of maps, in Section IX.
might have been augmented by others of the different kingdoms of
Europe or Asia ; but this was thought unnecessary, as, from the final
results of the operations before the student, he may very expeditiously
project a series of skeleton maps for the political divisions of Europe
or Asia, and form them into an atlas, to be filled up at his leisure.

182
This method of proceeding will give him a general knowledge of the
natural and artificial divisions of those two quarters of the earth's
surface, and make him well acquainted with the relative positions of
the individual countries, whose historical, political, civil, and; natural
geography, are to be properly learned from Pinkerton's "Geography."
Such a combination of plan in the study of this science, will give the
student a decided superiority, in rendering clear to his mind that
knowledge to be derived from application to books of voyages and
travels. But it isnot sufficient in maps to represent merely the situa
tion of places ; the connexions of countries, their extent, their divisions,
and their boundaries, are circumstances which belong to this part of
mathematical and political geography, as is likewise the form of the
terrestrial surface of those regions — that which is called the face qf the
country — that is to say, whether it is flat or mountainous, open or
wooded, dry or marshy. Means may be devised, sometimes pic
turesque, oftentimes arbitrary, to express upon trigonometrical surveys,
and topographical maps, these different circumstances, which, com
bined with the climate, and the meteorological phenomena of each
country, constitute its physical geography. Parts, more or less strongly
shaded, would represent declivities more or less steep, on which the
light loses itself in proportion as they are more perpendicular. In
topographical maps the mountains should be particularly regarded,
because the extent of forests in civilized countries having been con
siderably diminished, they have nearly disappeared from all maps;
but the inequalities of the ground, from the most lofty chains of moun-
to hills of the lowest order, should be expressed in a manner corre-,
sponding to all the geographical circumstances, and should, con
sequently, have a place in the details proportionate to their size.
Peaks, or insulated points in general, rest upon elevations more or
less considerable ; but the extent of which gives the outlines to deter
mine the form of the valleys, like the sinuosities of the coasts, which
are, with regard to the sea, like the hollows of mountains. If these.
remarks are well founded, the insulated points which usually mark
the mountains on the majority of niaps, must be very vague and
insignificant. Nothing is seen but that the country they occupy is
mountainous ; and, it has been suggested, that to write, here are moun
tains, would do as well ; for, either this mode ought to be adopted,
or we should indicate the course of their chains, their various de
pressions, and their connexions, either with each other, or with the
islands formed by the summits of the chains of sub-marine mountains.
In referring the reader to article 81, I present him with fine exercises

183
of this nature; as, from the immense chains of mountains which are"
there noticed, his topographical map, that would be projected for any
of the provinces or districts in which those fine chains of mountains
are situated, might-be formed to express in relief, though in an exag
gerated manner with respect to the scale of the parallels and meri
dians, so as to give an idea of the branches of the different mountains
connected with their perpetual inequalities and the depth of the valleys,
whether considered as elevated or depressed plains. Indeed, were a
topographical map composed in this style, you might indicate the
chains of mountains by the outline of their summits, to which might
be joined profiles or sections, following given lines, on which might
be constructed, from a convenient scale, the heights of the different
points on the face of the country. A profile or vertical section traced
in this way, Would exhibit at one glance the great features of a coun
try. Philippe Buache constructed a globe to shew the inequalities
on the terrestrial surface; and, in 1736, he traced, with particular care,
a section,- following the line which passes from Gape Tagrin to Rio
Grande, in which direction Africa and America approach the nearest
each- other. And Professor Leslie, (page 494), in his "Geometry,"
has combined and reduced, as a specimen of this operation, the sec
tions which the celebrated philosophic traveller Humboldt, has given
nf the continent of America, running in a twisted direction from
Acapulco to Vera Cruz, and connecting the Pacific with the Atlantic
Ocean. A ground plan expresses the several directions and distances
—a divided scale shews the horizontal distance in miles — the whole
mass is placed directly over this, and exterior of this mass, horizontal
parallels on a much larger scale, mark the elevation in feet; but as
the profile is really composed of four successive sections, denoted by
the ground plan, these sections are distinguished by opposite shadings.
The mensuration of heights by the barometer, is in most cases pre
ferred as the easiest, and often the more exact method ; and Leslie
has given (pages 490—500, " Elements of Geometry") an elegant
investigation of- Torricelli's immortal discovery of the weight of our
atmosphere, whence is determined the elevations of mountains. It is
from the 498th page to the end of the " Elements," that he gives a con
cise table of the most remarkable heights in different parts of the world,
expressed >in English feet. But it is La Croix who details the method
of M. Dussain Triel, as ingenious as satisfactory, to represent geome
trically the form of the surface of a. country, and I shall, therefore,
refer the reader to pages 71, 72, 73, and 74, of his elegant " Mathe-

184
matical and Critical Geography," prefixed as an introduction to Pin-
kerton's " Geography."
In looking over a map, one is struck with the strange appearance
of net-work, formed by the lines denoting the course of rivers — lines
which, like threads, intersect each other at every possible inclination,
from the most acute to the most obtuse angles.
Now, it is possible from the course of rivers and their branches, to
deduce, some general indications of the form of the ground in the dif
ferent countries of the earth. We know that the greatest rivers pass
through the greatest "valleys— that' torrents precipitate themselves over
rocks, and find their way, even if by a multiplicity pf windings,
through hilly grounds, till they join some more considerable stream,
which, in its turn, empties itself into some principal river, whose in
crease is proportionate to the prolongation of its course in approaching
nearer to the level of the sea. Retracing our steps* from the mouth
of a river to its source, as well as the streams from their confluence
with it, we shall generally arrive at the most elevated points. How
easy then, to infer the steepness of a declivity from the greater or less
degree of curvature iu the bed pf the river. In ascending from the
Indian, the Chinese, and Northern Seas, the great rivers of Asia, we
shall ultimately arrive in Thibet, and to the north of the country of
the Eleutheri ; from which it is obvious, that very high mountains sur?
round these countries, and here is found that plain referred to in arti
cle 80. It is very evident, from pursuing the course of the Asiatic
rivers, that, from this plain, three immense declivities extend them
selves to the above^eas, whatever may be the course of the auxiliary
streams, whose sources may be traced to branches of the intersecting
mountains. Note T. Page 134.
Very little sagacity is required to perceive that projections of this
kirid, preserving the relations arid superficial extent among different
countries, must have early and generally interested geographers ; for,
being not only easy of execution, from the concentric circles described
from a fixed point, taken as the axis of the map, and the meridians being
right lines terminating in that point, as in Problem X. or curved lines
as in the problem before us ; the quadrilaterals comprised between the
parallels and meridians are, as was mentioned in article 405, equiva
lent to those on the sphere. This projection is said to be much used
in France, and we know that Delille and D'Anville have employed it.

185
in fact, it is not surprising that this method should be popular: for,
assuming any right line as a radius, we have only to divide it into
nine equal parts, to take the ninth as our common centre of the con
centric parallels, divide the circle described by the lower extremity
of our radius into portions, equal to the axis meridian of the map,
and trace the other meridians by the laws of their decrease from
page 10th, and the work is done. See Ency. Brit. p. 541, method third.
A map of Europe projected on a very large scale, might combine all
the properties essential to a good map.
Though the methods of taking surveys belong properly to geometry
and trigonometry, as I have in the last Note mentioned topographical
maps, it will be necessary, before I conclude this note, to take some
notice of the manner in which general surveys are united in one
topographical plan or map. I am well aware, that many into whose
hands this volume shall come, would be more gratified, had I assigned
an entire section to the article of surveying, embracing, with trigono
metrical accuracy, the nice operations of geodaetical and maritime
surveying ; but it is to Colonel Mudge's " Trigonometrical Survey,"
the " Notes and Illustrations" of Leslie's " Elements," Hutton's " Ma-
matics," vol. 2d, Keith's ** Trigonometry," and Crocker's " Elements
of Laud Surveying," that I would refer the reader, for the information.
on this subject, which it is not possible to give in a note.
When an extensive geodaetical survey is taken, as, for example,
that of Britain, allowances must be made for the minute derangements
occasioned by the convexity i)f the surface of the earth.- In deter
mining the boundaries of that portion of the island south of Berwick
and Longton, near the Solway Frith, we might take Cary's Map of
England and Wales, and reduce the whole extent of country now
mentioned, to triangles and trapeziums ; for, by ascertaining the positions
of all the prominent objects within the scope of observation, we should
thence be able to measure the mutual distances and relative heights
of these objects, and, consequently, define the various contours which
mark the surface, and the number of acres it contains. In a survey of
this kind, the sinuosities of those portions of the land, over wliich the
grand lines of observation might pass in measuring along the coast,
would tolerably well compensate each other. But to determine the
Positions of remarkable headlands, and other conspicuous objects
that present themselves along the vicinity of a coast; as also the
situation of the various inlets, rocks, shallows, aud soundings, which
occur in approaching the shore, recourse must be had to the mixed
Jmethod of maritime surveying, in which we should have to. combine
o

186
a survey made on the land, with observations taken from boats moored
at proper intervals on the water.
When detached. surveys of this kind have been taken, the topo
graphical plan formed from them, in order1 to pass into the chorogra
phical map of England, described in Problem XII; of this Section,
we should not only assemble the different plans and survey's we had
made, but subject them to the projection oftlie meridians and paral*
lels traced in Fig. 3d, Plate XIX. ' But to do this with precision,
the student who has made himself pretty well acquainted' with mixed
mathematics, especially those principles which depend on geometry,
and its kindred sciences, trigonometry and perspective, will clearly
understand how to connect by one leading line, on his topographical
plan, those spaces comprised within triangles and trapeziums on the
leaves of the original survey, by fwlucing them to triangles and tra
peziums formed on the topographical plan, like those on the sheets of
the survey ; so that the sides of the first may be to those of the second,
in the relation exacted by the reduction. And it is very obvious,
that, in this procedure, the method of reduction by dividing each of
the corresponding parts of the whole, into divided squares (Note LJ
p. 177,) is the most convenient for the construction of the details of
the country; and that, the more those squares are multiplied, pro
gressively is the facility in judging of the place to be occupied in each
square, by the points and circumstances contained within it ; and
inscribing them with a strict resemblance in the correspondent squares;
traced on the reduced plan. But as these squares might not corre
spond strictly with the quadrilaterals1 on the map before us, we should
take, by reference to the sides of the first, that is to say, the sides of
the parallels and meridians on the topographical plan; the distances
of the principal points therein contained, and convert these distances
into subdivisions of the degrees of latitude and longitude, and the like
are taken from the parallel and meridian contiguous to. the correspond*
ing quadrilaterals on the map. •> . . ...
N,ote 11, Page 133.
This construction is hinted at in Goldsmith's " Popular Geography^"
and a skeleton map is given, but which is perfectly Unintelligible to
every beginner, as there is no ready 'elucidation of the principle on
which the centre of all the concentric parallels is- posited at thirty de
grees beyond the pole, and in the axis of the map. But as the old work,-
from which Goldsmith copied that map, did not furnish him with thi?

187
accessary assistance, his readers must beat it out of their own brains,
as well as they can, or abandon the work, from its resemblance to the
eternal labour of the life-loving Sisyphus. Were w,» to adopt De Lisle's
projection' of a map of Asia, according to the polar method, the equa
tor being circular, the degrees of longitude would thereby, in all the
parallels south of the equator, be greater than those on the equator,
and those on the equator, less than the degrees of latitude with which
they should be equal. Note V. Page 144.
There is to be found in the " Encyclopaedia Britannica," a diagram
of an Analemma, of which the figure I 'have given is similar, and in
page 544, article 123, fifth edition, there is described its construction;
Girt the manner In which that description is wrapt up, makes it intel
ligible only to those who are well acquainted with the laws of the
orthographic' projection, and when the student shall have examined
the one before him, with that in the Encyclopaedia, he will determine
which merits the preference. Mr. Planta, with a disinterestedness
which distinguishes. him as a librarian, gave me every" facility of-access.
to- the work of Ptolemy I have referred to, and which I could .only.
find in the British Museum.

FINIS.

Page 23d— Insert an asterisk * after the numerals 1380 *, to correspond with
the reference made at the bottom of the page.
 '86th— In the fourth line from the top, for (101), read (83).
— — 28th— In the fourteenth line from the top, for (89), read (83).
— — 49th — In the foarteenth line from the top, destroy the comma betweea
os and is, and read " as is."
•— — lOlst-r-In the fourth line from the top, for (356), read (355).
•«¦— 166th— >Ib the eighth line from the bottom, transfer the comma to the
left of the word aimrters, and read " quarters which," &c