Abel Flint.

A system of geometry and trigonometry, together with a treatise on surveying .. online

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HARTFORD : < ^ '.


District o/ Connecticut, *.

BE IT REMEMBERED, That ou the twenty-fifth day of September,
in the forty-third year of the Independence of the United States of
America, Oliver D. Cooke & Co., of the said district, have deposited in this
office the title of a hook, the right whereof they claim as Proprietors, in the
words folio-wing, to wit

" A System of Geometry and Trigonometry : together with a Treatise on
Surveying ; Teaching various ways of taking the Surrey of a Field ; Also to
Protract the same and find the Area. Likewise, Rectangular Surveying ;
or, an Accurate Method of calculating the Area of any Field Arithmetically
without the necessity of Plotting it. To the whole are added several
Mathematical Tables, necessary for solving Questions in Trigonometry and
Surveying ; with a particular explanation of those Tables, and the Manner
-of using them. Compiled from various Authors. By Abel Flint, A. M.
Fifth Edition, with important Additions, By George Gillet, Surveyor General
of the State of Connecticut."

" In conformity to the Act of the Congress of the United States, entitled
" An Act for the encouragement of Learning, by securing the copies of
Maps, Charts, and Books, to the authors and proprietors of such copies,
during the times therein mentioned."

Clerk of the District of Connecticut.
A true copy of Record, examined and sealed by me,

Clerk of the District of Connecticut.

* * *,


HAVING perused, with some attention, the following'
Treatise on Surveying, in Manuscript, it appears to me
to be estimable for its simplicity and perspicuity ; and,
by excluding all matter but remotely connected with
the main subject, and reducing the Tables of Log-
arithms, of Logarithmic Sines, Tangents, and Secants,
and of Difference of Latitude and Departure, without
impairing their use, in their application to most cases
which occur in common Surveying, and supplying any
possible defect by a Table of Natural Sines, to com-
prise, in the limits of a pocket volume, whatever is
most essential and most useful in the Art, including the
important modern improvement of RECTANGULAR SUR-
VEYING ; and on the whole, particularly from the size
of the volume, to be well adapted to general use.


FARMINGTON, Sept 20tb, 1804.


WE the subscribers have carefully perused a Trea-
tise on Surveying, prepared for the Press* by the Rev.
Abel Flint of Hartford ; and find it worthy of the pub-
lic patronage. Every thing not immediately necessarj
for the practical Surveyor has been excluded ; while it
comprises all which is requisite in Field Surveying, both
on the old and new plan ; elucidated and explained with
a degree of conciseness and perspicuity not usually to
be found in Treatises on the same subject. The Ma-
thematical Tables are reduced to less than half the size
occupied by others ; and any inconvenience which
might result from such reduction, is^ obviated by the in-
sertion of a Table of Natural Sines, not usually found
in works of this nature. The Surveyor who shall own
this will not be under the necessity of purchasing GIB-
SON, which is a more expensive work.

ASHER MILLER, Surveyor General.
GEORGE GILLET, Deputy Surveyor
for Tolland County,

MlDtLETOWN, Oct. 3, 1804.


THE following work is chiefly a compilation from
other Books ; and but very little new is added, ex-
cept a more full explanation, than has yet been
published, of RECTANGULAR SURVEYING, or the
method of calculating the Area of Fields arithme-
tically, without drawing a plot of them and mea-
suring with a Scale and Dividers, as has been the
common practice > and also a more particular ex-
planation of the use of Natural Sines than is con-
tained in most Mathematical Books.

The Compiler has endeavoured to render this
work so easy and intelligible that a Learner Will
require but little assistance from an Instructer, ex-
cept with regard to the construction and use of
Mathematical and Survey ing Instruments. Before,
however, he eaters, on the study of this Book he
must be well acquainted with common Arithmetic,
with PecimaL Fractions, and the Square Root;
and he must also know the various characters or
marks used in, Arithmetic,

A Surveyor will doubtless find many questions
arise in the course of his practice, for the solution
of which, no particular directions are here given ;
nor is it possible to give directions for every case
that may occur. In all practical Sciences much
must be left to the judgment of the practitioner,
who, if he is well acquainted with the general
principles of his Art, will readily leara to apply
those principles to particular cases.

The primary design of this treatise is to teach
common Field Surveying ; at the same time it con-
tains the elements of Surveying upon a larger


scale ;'and the system of Geometry and Trigono-
metry with which it is introduced, with the Prob-
lems for the mensuration of Superficies, as also the
Mathematical Tables at the end, will be found use-
ful for many other purposes. It would be well,
therefore, for those who do not intend to become
practical Surveyors to acquaint themselves with
what is here taught ; and with this view the follow-
ing work is very proper to be introduced into Aca-
demies, and those higher Schools which are design-
ed to fit young men for active business in life. In-
deed every person who frequently buys and sells
land should learn to calculate the Contents of a
field arithmetically ; a knowledge which may be
acquired in a very little time, from the particular
explanation here given of that method.

Notwithstanding; the many Books already pub-
lished on the subjects here treated upon, it was
thought a work of this kind was really wanted, and
that if judiciously executed it would be useful. It
is more particularly necessary at the present time
in Connecticut, as the Legislature of the State
have lately enacted a Law on the subject of Sur-
veying, in consequence of which more attention
must be paid to the Theory of that Art than has
been common.

These considerations induced the Compiler to
select from various publications what appeared to
him important; and to arrange the whole in a
method best adapted, in his view, for teaching that
useful Art. How far he has succeeded in his en-
deavours to simplify the subject, and render it easy
to the Learner, must be submitted to the test of

HARTFORD, Conn. October, 1804.


THE System of Geometry is divided into two
parts. The first contains Geometrical Definitions
respecting Lines, Angles, Superficies, &c. The
second part contains a number of Geometrical
Problems necessary for Trigonometry and Survey-

The System of Trigonometry is also divided
into two parts : and teaches the solution of ques-
tions in Right and Oblique angled Trigonometry,
by Logarithms and also by Natural Sines.

The Treatise on Surveying is divided into three
parts. Part first treats of measuring Land, and is
divided into three Sections. The first contains
several Problems respecting Mensuration, and for
finding the Area of various Right-lined Figures
and Circles.

The second Section teaches different methods
of taking the Survey of Fields ; also to protract
them, and find their Area in the manner commonly
practised, and likewise by Arithmetical and Tri-
gonometrical calculations, without measuring Di-
agonals and Perpendiculars with a Scale and Di-
viders ; interspersed with sundry useful rules and

The third Section is a particular explanation
and demonstration of Rectangular Surveying, or
the method of computing the Area of Fields from
the Field Notes, by Mathematical Tables, without
the necessity of plotting the Field. To this Sec-
tion is added a useful Problem for ascertaining the
true Area of a Field which has been measured by
a Chain too long or too short.


Part second treats of laying out Land in various

Part third contains sundry Problems and Rules
for dividing Land and determining the true Course
and Distance of dividing Lines, or from one part
of a Field to another. To this is added an Appen-
dix concerning the Variation of the Compass and
Attraction of the Needle ; also, a rule to find the
difference between the present Variation, and that
at a time when a Tract was formerly surveyed, in
order to trace or run out the original lines.

The Mathematical Tables, are a Traverse Ta-
ble, or Table of Difference of Latitude and De-
parture, calculated for every Degree and quarter
of a Degree, and for any distance up to 50 ; a Ta-
ble of Natural Sines calculated for every Minute ;
a Table of Logarithms comprised in four pages,
yet sufficiently extensive for common use ; and a
Table of Logarithmic or Artificial Sines, Tangents,
and Secants, calculated for every 5 Minutes of a
Degree. To these Tables are prefixed particular
explanations of the manner of using them.


GEOMETRY is a Science which treats of the properties of


Geometrical Definitions. +

1. A Point is a small Dot; or, Mathematically considered, is
fhat which has no parts, being of itself indivisible.

2. A Line has length but no breadth.

3. A Superficies or Surface, called also Area, has length and
breadth, but no thickness.

4. A Solid has length, breadth, and thickness.

5. A Right Line is the shortest that can be drawn between
two Points.

Fig. I. ,

G. The inclination of two Lines meeting
oue another, or the opening between them,
is called an Angle. Thus at B. Fig. 1. is an
Angle, formed by the meeting of the Lines
AB and BC.

7. If a right Line CD. Fig. 2. fall upon an-
other Right Line AB, so as to incline to nei-
ther side, but make the Angles on each side
equal, then those Angles are called Right
Angles ; and the Line CD is said to be Per-
pendicular to the other Line.



Fig. 3,

8. An Obtuse Angle is greater than a Right
Angle; as ADE. Fig. 3.

9. An Acute Angle is less than a Right
Angle; as EDB. Fig. 3.

Note. When three letters are used to express aa Angle, thr
middle letter denotes the angular Point.

Fig. 4.

10. A Circle is a round Figure, bounded
by a Line equally distant from some Point,
which is called the Centre. Fig. 4.

11. The Circumference or Periphery of
a Circle is the bounding Line ; as ADEB.
Fig. 4.

12. The Radius of a Circle is a Line
drawn from the Centre to the Circumference ;
as CB. Fig. 4. Therefore all Radii of the
same Circle are equaK

13. The Diameter of a Circle is a Right
Line drawn from one side of the Circumfer-
ence to the other, passing through the Centre ;
and it divides the Circle into two equal parts,
called Semicircles ; as AB or DE. Fig. 5.

14. The Circumference of every Circle is
supposed to be divided into 360 equal part?,
called Degrees ; and each Degree into 60
equal parts, called Minutes ; and each Minute

into 60 equal parts, called Seconds ; and these into Thirds, &c.

JVote. Since all Circles are divided into the same number ot
Degrees, a Degree is not to be accounted a quantity of any
determinate length, as so many inches or feet, &c. but is
always to be reckoned as being the 360th part of the Cir~
cumference of any Circle, without regarding the bigness-
of the Circle.

15. An Arch or Arc of a Circle is any part of the Circumfer-
euce ; as BF or FD. Fig. 5 ; and is said to be an Arch of so
many Degrees as it contains pacts of 360 into which the whok
Circle is divided.



16. A Chord is a Right Line drawn from
one end of an Arch to the other, and is the
measure of the Arch ; as HG is the Chord of
the Arch HIG. Fig. 6.

Fig. 7.

. The Chord of an Arch of 60 degrees is equal in length
to the Radius of the Circle of which the Arch is a part.

17. The Segment of a Circle is a part of a Circle, cut off by
a Chord ; thus the^pace comprehended between the Arch HIG
and the Chord HG is called a Segment. Fig. 6.

18. A Quadrant is one quarter of a Circle ; as ACB. Fig. 6.

19. A Sector of a Circle is a space contained between two
Radii and an Arch less than a Semicircle ; asBCDorACD,
Fig. 6.

20. The Sine of an Arch is a Line drawn
from one end of the Arch, perpendicular to
the Radius or Diameter drawn through the
other end : Or, it is half the Chord of double
the Arch ; thus HL is the Sine of the Arch
HB. Fig. 7.

21. The Sines on the same Diameter in- A
crease in length till they come to the Centre,
and so become the Radius. Hence it is plain
that the Radius CD Fig. 7. is the greatest
possible Sine, or Sine of 90 Degrees.

22. The Versed Sine of an Arch is that part of the Diameter
or Radius which is between the Sine and the Circumference ?
thus LB is the Versed Sine of the Arch HB. Fig. 7.

23. The Tangent of an Arch is a Right Line touching the
Circumference, and drawn perpendicular to the Diameter ; ami
is terminated by a Line drawn from the Centre through the
other end of the Arch ; thus BK is the Tangent of the Arch
BH. Fig. 7.

JVote. The Tangent of an Arch of 45 Degrees is equal in
length to the Radius of the Circle of which the Arch i^a

. The Secant of an Arch is a Line drawn from the Centre


through one end of the Arch till it meets the Tangent ; thus-
CK is the Secant of the Arch BH. Fig. 7.

25. The Complement of an Arch is what the Arch wants of
90 Degrees, or a Quadrant ; thus HD is the Complement of the
Arch BH. Fig. 7. ^

26. The Supplement of an Arch is what the Arch wants
of 180 Degrees, or a Semicircle ; thus ADHis the Supplement
of the Arch BH. Fig. 7.

27. The Sine, Tangent or Sejant of the Complement of any
Arch is called the Co-Sine, Co-Tangent, or Co-Secant of the
Arch ; thus, FH is the Sine, Dl the Tangent, and Ci the Secant
of the Arch DH ; or they are the Co-Sine, Co-Tangent, and Co-
Secant of the Arch BH. Fig-. 7.

28. The measure of an Angle is the Arch of a Circle con-
tained between the two Lines which form me angle, the angular
Point being the Centre ; thus, the Angle HCB. Fig. 7. is mea-
sured by the Arch BH : and is said to contain so many Degrees
m the Arch does.

Note. An Angle is esteemed greater or less according to the
opening of the Lines which form it, or as the Arch inter^
cepted by those Lines contains more or fewer Degrees.
Hence it may be observed, that the bigness of an Angle
does not depend at all upon the length of the including
Lines ; for all Arches described on the same Point, and
intercepted by the same Right Lines, contain exactly the
same number of Degrees, whether the Radius be longer or

29. The Sine, Tangent, or Secant of an Arch is also the Sine ;
Tangent, or Secant of the Angle whose measure the. Arch is.

Fi. 8.

30. Parallel Lines are such as are equally -&, - B
distant from each other ; as AB and CD.

Fig. 8r

31. A Triangle is a Figure bounded by
rhree Lines ; as ABC. Fig. 9.

32. An Equilateral Triangle has its three
aides equal in length to each other. Fig. 9.


33. An Isocles Triangle has two of its sides
equal, and the other longer or shorter. Fig. 10.

Fig. 11.

34. A Scalene Triangle has three unequal
sides. Fig. 11.

35. A Right Angled Triangle has one
Right Angle. Fig. 12.

Fig. 12.

Fig. 13.

36. An Obtuse Angled Triangle has one
Obtuse Angle. Fig. 13.

37. An Acute Angled Triangle has all its Angles Acute,
Fig. 9, er 10.

38. Acute and Obtuse Angled Triangles are called Oblique
Angled Triangles, or simply Oblique Triangles ; in which the
bottom Side is generally called the Base and the other two.

39. In a Right Angled Triangle the longest side is called the
Hypothenuse, and the other two, Legs, or Base and Perpen-

Note. The three Angles of every Triangle being added to-
gether will amount to 180 Degrees ; consequently the two
Acute Angles of a Right Angled Triangle amount to 90
Degrees, the Right Angle being also 90*



40. The perpendicular height of a Trfen-
gle is a Line drawn from one of the Angles
to its opposite side ; thus, the dotted Line
AD. Fig. 14. is the perpendicular height of
the Triangle ABC.

JVbte. This Perpendicular may be drawn from either of the
Angles ; and whether it falls within the Triangle, or on
one of the Lines continued beyond the Triangle, is imma-

4K A Square is a Figure bounded by four
equal sides, and containing four Right Angles.
Fig. 15.

42. A Parallelogram, or Oblong Square, is
a Figure bounded by four sides, the opposite
ones being equal and the Angles Right. Fig.

Fig. 15.

Fig. 16.

Fig. 17.

43. A Rhombus is a Figure bounded by
four equal sides, but has its Angles Oblique.
Fig. 17.


Fig. 18.

44. A Rhomboides is a Figure bound-
ed by four sides, the opposite ones being
equal, but the Angles Oblique. Fig. 18.


45. The perpendicular height of a Rhombus or Rhomboides
is a Line drawn from one of the Angles to its opposite side :
thus, the dotted Lines AB. Fig. 17. and Fig. 18. represent the
perpendicular height of the Rhombus and Rhomboides.

Fig. 19.

46. A Trapezoid is a Figure bounded by
lour sides, two of which are parallel though
of unequal lengths. Fig. 19. and Fig. 20.


Fig. 20.

Note. Fig. 19. is sometimes called a Right Angled Trape-

Fig. 21,

47. A Trapezium is a figure bounded by
four unequal sides. Fig. 21.

48. A Diagonal is a Line drawn between
two opposite Angles ; as the Line AB. Fig.

49. Figures which consist of more than four sides are called
Polygons ; if the sides are equal to each other they are called
regular Polygons, and are sometimes named from the number
of their sides, as Pentagon, or Hexagon, a Figure of five or six
sides, &c. ; if the sides are unequal, they are called irregulav


Geometrical Problems.

PROBLEM I. To draw a Line parallel to
another Line at any given distance ; as at the > ;. - "-^
Point D, to make a Line, parallel to the Line

Fig. 22.

Fig. 22,




With the Dividers take the nearest distance between the
Point D and the given Line AB ; with that distance set one
foot of the Dividers any where on the Line AB, as at E, and
draw the Arch C ? through the Point D draw a Line so as just
to touch the top of the Arch C.

A more convenient way to draw parallel Lines is with ;i
parallel Rule. :-fc&V1

Fig. 23.

PROBLEM II. To bisect a given Line;
or to find the middle of it. Fig. 23.


Open the Dividers to any convenient distance, more than
half the given Line AB, and with one foot in A, describe an
Arch above and below the Line, as at C and D ; with the same
distance, -and one foot in B, describe Arches to cross the former ;
lay a Rule from C to D, and where the Rule crosses the Line,
as at E, will be the middle.

Fig. 24.

PROBLEM III. To erect a Perpendicular
from the end, or any part of a given Line. Fig.


Open the Dividers to any convenient distance, as from D to
A, and with one foot on the Point D, from which the Perpen-
dicular is to be erected, describe an Arch, as AEG ; set off the
same distance from A to E, and from E to G ; upon E and G
describe two Arches to intersect each other at H ; draw a
Line from H to D, and one Line will be perpendicular to the

JVbfe. There are other methods of erecting a Perpendicular,
but this is the most simple.


Fig. 25.

PROBLEM IV. From a given Point,
as at (', to drop a Perpendicular on a&.
given Line AB. Fig. 25.

With one foot of the dividers in C describe an Arch to cut
the given Line in two places, as at F and G ; upon F and G
describe two Arches to intersect each other below the Line as
at D ; lay a Rule from C to D and draw a Line from C to the
given Line.

Perpendiculars may be more readily raised and let fall, by a
small Square made of brass, ivory, or wood.

Fig. 26.

PROBLEM V. To make an Angle at
*qiial to a given Angle ABC. Fig. 26.


Open the dividers to any convenient distance, and with one
foot in B describe the Arch FG ; with the same distance and
one foot in E, describe an Arch from H ; measure the Arch FG,
and lay off the same distance on the Arch from H to I ; draw
a Line through I to E, and the Angles will be equal.

Fig. 27.

PROBLEM VI. To make an Acute Jingle f
equal to a given number of Degrees, suppose f
36. Fig. 27. I


Draw the Line AB to any convenient length ; from a Scale
of Chords take 60 Degrees with the dividers, and with one foot
in. B describe an Arch from the Line AB ; from the same Scale
take the given number of Degrees, 36, and lay it on the Arch
from C to D ; draw a line from B through D, and the Angle at
B will be an Angle of 36 Degrees.



Fig. 26,

PROBLEM VII. To make an Obtuse Jin- " *
'c, suppose of 110 Degrees. Fig. 28.


Take a Chord of 60 Degrees as before, and describe an Arch
greater than a Quadrant ; set off 90 Degrees from B to C, and
from C to E set off the excess above 90, which is 20 ; draw a
Line from G through E, and the Angle will contain 110 De-

Note. In a similar manner Angles may be measured ; that is.
with a Chord of 60 Degrees describe an Arch on the an-
gular Point, and on a Scale of Chords measure the Arch
intercepted by the Lines forming the angle.

A more convenient method of making and measuring Angles
is to use a Protractor instead of a Scale and Dividers.

Fig. 29.

B ^ O


PROBLEM VIII. To make a Triangle oj
three given Lines, as BO, BL, LO. Fig. 29.

Draw the Line BL from B to L ; from B, with the length of
the Line BO, describe an Arch as at O ; from L, with the
length of the Line LO, describe another Arch to intersect the
former ; from O draw the Lines OB and OL, and BOL will be
the Triangle required.

Fig. 30.

PROBLEM IX. To make a Right Jingled
Triangle, the Hypothenuse and Jingles being
hen. Fig. 30.

Suppose the Hypothenuse CA 25 Rods or Chains, the angl*



at C 35 3Q' and consequently the Angle at A 54 30'.
Note after the 39th Geometrical Definition.


Note. When degrees and minutes are expressed, they are
distinguished from each other by a small cipher at the
right hand of the degrees, and a dash at the right hand of
the minutes ; thus 35 30' is 35 degrees and 30 minutes.

Draw the Line CB an indefinite length ; at C make an Angle
of 35 30' ; through where that number of Degrees cuts the
Arch draw the Line CA 25 Rods, which must be taken from
some Scale of equal parts ; drop a Perpendicular from A to B,
and the Triangle will be completed.

Note. The length of the two Legs may be found by mea-
suring them upon the same scale of equal parts from which
the Hypothenuse was taken.

PROBLEM X. To make a Right Jin-
gled Triangle, the Jingles and one Leg being
given. Fig. 31.



Suppose the Angle at C 33 15', and the Leg AC 285.

Draw the Leg AC making it in length 285 ; at A erect a Per-
pendicular an indefinite length ; at C make an Angle of 33
15'; through where that number of Degrees cuts the Arch,
draw a Line till it meets the Perpendicular at B.

Note. If the given Line CA should not be so long as the
Chord of 60, it may be continued beyond A, for the pur-
pose of making the Angle.

Fig. 32.


PROBLEM XI. To make a Right Angled
Triangle, the Hypothenuse and one Leg being
given. Fig. 32.

Suppose the Hypothenuse AC 40, and the Leg AB 28.

Draw the Leg AB in length 28 ; from B erect a Perpendi-

qular an indefinite length ; take 40 in the Dividers, and setting


one foot in A, wherever the other foot strikes the Perpendicular

Online LibraryAbel FlintA system of geometry and trigonometry, together with a treatise on surveying .. → online text (page 1 of 28)