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William Shepherd.

Systematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) online

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bers, in which state it contained the logarithms of all numbers
from 1 to 100,000 to ten places of decimals, together with a
table of logarithmic sines, tangents and secants, to the same ex*
tent, for every minute of the quadrant.

Mr. Briggs completed a table of logarithmic sines and tan-

VOL. I. 2 I



482 MATHEMATICS.

gents for the luindredth part of every degree, to fourteen
places of decitnuls, together with a table of natural sines for
the same parts to fifteen places, and the tangents and secants
of the same to ten places, with an account of the construction
of the whole; which work was likewise printed at Gonda by
Valcq, in 16S3, and upon his death a preface to it was sup-
plied by Mr. Henry Gellibraud, who added to it the applica-
tion of logarithms to plane and spherical trigonometry, which
he published in the same year under the title of " Trigono-
metry Britannica." Valcq at the same time published his own
great work, entitled " Trigonometria Artiticialis," which con-
tains the logarithmic sines and tangents of every ten seconds of
the quadrant, to ten places of figures, besides the index, and
the logarithms of the first twenty thousand numbers, to the
same number of places, with the differences of each ; the
whole being preceded by a full description of the tables, and
the application of them to some of the principal problems in
plane and spherical Trigonometry. Several smaller tables of
these logarithms were also published about this time by Gunter,
Wingate, and others. Gunter, it should be noticed, first ap-
plied the logarithms of numbers, together with those of the
sujes and tangents, to a ruler, in the form of a two foot scale,
which is still known by his name ; by which proportions in tri-
gonometry, navigation, and other subjects, may be performed,
to a degree of accuracy sufficient for many practical purposes,
by the mere application of a pair of compasses.

The common logarithmic (^non was first reduced to its
most convenient form by Mr. John Newton, in hb ** Trigo-
nometria Britannica," printed in 1658, which contains the
logarithms of the first 100,000 numbers to 8 places of deci-
mals, arranged in the same manner as they are in our table*
at present, as also the logarithmic sines and tangents to the
same exteut. The greater part of these tables are superse-
ded by those of a later date, some of which we shall have
occasion to notice in the sequel.

Trigonometry, or the resohition of triangles, is founded



TRIGONOMETRY. 4^

tipon the mutual proportions which subsist between the sides
and angles of triangles, which proportions are kuDwn by
tinding the relations between the radius of a circle, and certain
other lines drawn in and about the circle, denominated chords,
sines, tangents, secants, 8cc. We have seen that the ancients
performed their trigonometry by means of chords, which have
been long since abandoned for the use of sines, tangents, and
secants.

The proportion of the sines, tangents, &c to their radius, is
sometimes expressed in common or natural numbers, which
constitute what are called the tables of natural sines, tan*
gents and secants. Sometimes it is expressed in logarithms,
being the logarithms of the natural sines, tangents, 8cc. These
last constitute the table of artificial sines, tangents, secants,
&c. : and sometimes the proportion is not expressed in numbers :
but the several sines, tangents, &c. are actually laid down upon
certain scales. In trigonometry, as the angles are measured
by arcs of a circle described about the angular point, so the
whole circumference of the circle is supposed to be divided
into 360 parts, called degrees, and each degree is divide dinto
60 parts, called minutes, and each minute, into 60 seconds ;
sometimes by a similar division, the seconds are divided into
thirds^ and so on. An angle therefore is said to consist of so
many degrees, minutes and seconds, as are contained in the
arc that measures the angle, or that is intercepted between the
sides or legs of the angle.

The sines, tangents, secants, co-sines, 8cc. of every degree,
minute, second, &c. of a quadrant, are calculated to the radius of
1, and ranged in tables for use, as also the logarithms of the
same ; forming the triangular canon.

There are two methods used in investigating the principles
of trigonometry : viz. by geometry and algebra. By the
former, the various relations of the sines, co-sines, tangents
of arcs or angles, and those of the sides and angles of
triangles, are deduced immediately from the figuix-s to which
the several inquiries are referred, each particular case re-

2 I 2



484 MATHEMATICS.

quiring its own method, and resting on evidence peculiar io
itself. By tlie algebraical method, the nature and properties
of the linear-angular quantities being first defined, some
general relation of these quantities, or of them in connexion
with a triangle, is expressed by one or more equations, and
then other theorems of use iu this branch of science, are
developed by the simple reduction of the first equation. Thus
it is observed, " the rules for the three fundamental cases in
plane trigonometr}', which may be deduced by three indepen-
dent geometrical investigations, are obtained algebraically, by
forming between the three data, and the three unknown quan-
tities, three equations, and obtaining, in expressions of known
terras, the value of each of the unknown quantities, the others
being exterminated by the usual processes. Each of these
methods has its peculiar advantages. The geometrical method,
in this, as in all other cases, carries conviction at every step ;
and, by keeping the objects of inquiry constantly before the
eye, serves to guard against the admission of error. The
algebraical method, on the contrary, requiring little aid
from first principles, excepting at the conmieucement of the
process, is rather mechanical than mental, and requires fre-
quent checks to prevent any deviation from the truth. The
geometrical method is direct and rapid, in producing the con-
clusions at the outset of trigonometrical science j but slow
and circuitous in arriving at those results which the modem
state of science requires : while the algebraical method,
though sometimes circuitous iu the developement of the
mere elementary theorems, is very rapid and fertile in pro-
ducing those curious and interesting formulse, which are
wanted in the higher branches of pure analysis, in mixed
mathematics, and more particularly in physical astronomy."

in practice, three things, in every plane triangle, must be
given to find the rest : and of these three, one, at least, must
be a iride. There are three cases which include all the varieties
tliat can happen: (1) When two of the three given things
are a side, and its opposite angle. (2) When two sides and



TRIGONOMETRY. 48^

their included angle are given. (3) When the three sides are
given. Each of these cases may be resolved in three differetit
ways, viz. by geometrical construction by arithmetical com-
putation or instrumentally. In resorting to the first of these
methods, the student will find himself well prepared for the
pursuit, if he has before attended to the Geometrical construc-
tion of Problems, of which there are many excellent exam-
ples in the latter part of Mr. Thomas Simpson's Geometry.
In this method, the triangle is constructed by laying down the
sides from a scale of equal parts, and the angles from a scale
of chords, or a protractor, and then measuring the unknown
parts by the same scale or instrument from which the others
were taken.

In the mode by arithmetical computation ; having stated
the proportion according to the proper rule, multiply the se-
cond and third terms together, and divide the product by the
first, and the quotient will be the fourth term required ; that
is, when the natural numbers are used. In working by loga-
rithms, the logarithms of the second and third terms are to be
added together, and from the sum, the logarithm of the first
term is to be taken, and the number answering to the re-
mainder, will be the fourth term sought.

In the third method, or instrumentally, as by logarithmic
lines, on one side of a common two-foot scale, the legs of the
compasses are to be extended from the first term to the se-
cond or third, as they may happen to be of the same kind,
and that extent will reach from the other terra to the required
fourth term, taking both extents towards the same end of the
scale.

Among the many works on trigonometry that may be re-
commended to the student's attention, the following require to
be noticed.

" The Trigonometer's Guide, by Benjamin Martin," in two
volumes, 8vo. In the course of this work, the examples are
very pumerous, and applied to the various departments in n^r



486 MATHEMATICS.

tural pliilosophy. They are worked according to the three
methods mentioned above.

On nearly the same plan, but on a much more contracted
scale, is a small work, entitled " An Easy Introduction to
Plane Trigonometry : the application of it to the measuring
Heights and Distances ; to the several branches of Natural
Philosophy, Land Surveying, 8cc. by C. Ashworlh, D. D."
This treatise is practical, and, like the last, adapted for per-
sons wlio have not gone deeply into mathematical studies. It
contains a great number of examples, and, being of a small
price, may be used in connexion with those which are more
theoretical, and some of which are very scanty in their exem-
plifications.

^ " The Elements of Plane and Spherical Trigonometry," by
Dr. Robert Simson, consist of only about forty pages, and
are attached to his Elements of Euclid. This treatise con-
tains the elementary part, and is, says the author, " much
the same as the common treatises, excepthig that the demon-
strations of several of the propositions are changed into others
that were thought better."

" Trigonometry, Plane and Spherical, with the construc-
tion and application of Logarithms." By Thomas Simpson,
F. R S. This, like the last, is brief, but, for the most part,
clear and perspicuous. The section on the nature and con-
struction of logarithms, with their application to the doctrine
of triangles, is short, but satisfactory. The author having ex-
hibited the maimer of resolving all the common cases of plane
and spherical triangles, he subjoins a few propositions for the
solution of the more difficult cases which sometimes occur.

A larger and somewhat plainer disquisition on the nature
and use of logarithms stands as an introduction to " A Trea-
tise on Plane and Spherical Trigonometry," by the Rev.
S. Vince, now Doctor Vince. This is intended as part of a
mathematical course, recommended to the students at Cam-
bridge, ami is therefore adapted particularly to the viewp of



TRIGONOMETRY. 48?

the professors of that learned University ; it may, however,
answer all the purposes of other persons who are in pursuit of
trigonometrical knowledge.

The " Elements of Plane Trigonometry" are neatly and
clearly stated by Mr. Leslie in his work on Geometry, of
which we have before spoken ; and to which we mean to
make ourselves indebted for certain illustrations on several of
the branches of science flowing from the principles of trigo-
nometry. As an introduction to this little piece, the learned
professor observes, the sides of a triangle are measured, by
referring them to some definite portion of linear extent, which
is fixed by convention. The mensuration of the angles is
effected by means of that universal standard derived from the
partition of a circuit. Since angles are proportional to the
intercepted arcs of a circle described from their vertex, the
subdivision of the circumference, therefore, determines their
magnitude. A quadrant, as it corresponds to a right angle,
hence forms the basis of angular measures ; but these mea*
sures depending on the relation of certain orders of lines con-
nected with the circle, he sets out with previously investi^
gating these; and, in order the more clearly to discern the
connexion of the lines derived from the circle, Mr. Leslie
traces their successive values, while the corresponding arc is
supposed to increase. Having thus taken his preliminary
steps, his whole tract is comprised in a few theorems and pro-
blems, which will not present any serious obstacles to the
young mathematician, if he be well grounded in the principles
of geometry. This, in mathematics, is, what JDemosthenes
said, pronunciation was to oratory, the first, the second, and,
in short, every thing.

" Elements of Trigonometry, plane and spherical, applied
to the most useful problems in heights, distances. Astronomy
and Navigation," By William Payne, author of Elements of
Geometry, already referred to, is a very easy and useful
treatise adapted to the use of learners. -'''

Mr. Woodhouse's Treatise on Plane and Spherical Trigo^



488 MATHEMATICS.

nometry, may be recommended ; but it b less easy than most
of those already noticed.

There are, no doubt, other tracts of the smaller kind, of
real value on trigonometry, of which our own knowledge does
not allow us to speak ; and such, we are persuaded, is that by
Mr. Bridge. We shall close our account by mentioning two,
which are carried to a much greater extent than any of
those of which we have already spoken.

*' A Treatise on Plane and Spherical Trigonometry, with
their most considerable applications." By John Bunnycastle.

"An Introduction to the Theory and Practice of Plane
pnd Spherical Trigonometry, and the Stereographic Projec-
tion of the Sphere, including the Theory of Navigation," &c.
By Thomas Keith.

To Air. Bonnycastle's introduction, the historical part of
this chapter in our work is much indebted, he having more
neatly and concisely stated the rise and origin of the trigono-
metrical and logarithmic calculus than any other author in our
recollection. The author, as will be seen in the foregoing
pages, has attributed to the respective authors the honour due
to them on account of their several discoveries. The work
itself is furnished with a great variety of rules and examples ;
and, in the second edition, which is much enlarged, nearly all
the practical questions that had been before given, and many
of which, he says, had passed in the same state from one
work to another for more than a century ; he has re-proposed,
with new data, " and in every case," the author adds, " where
it was judged necessary, both the examples, and the rules
upon which they depend, have been carefully attended to, and
reduced to their most commodious and approved forms."

" The Introduction," by Mr. Keith, tliough not lai^fer in
the number of its pages, by its mode of printing, contains a
deal more matter, perhaps double the quantity. The author
means it for a complete treatise in itself; and, in answer to
those who may suppose it too much extended for an introduc-
tory treatise, he says, " llie student is here supplied with three



TRIGONOMETRY, 489

distinct treatises, which are all essential towards his future pro-
gress in the science : 1st. A Treatise on Logarithms, and the
necessary Tables ; 2d. A Treatise on Practical Trigonometry ;
and 3d. A Theoretical Treatise on the Subject." In the part of
the volume which comprehends Spherical Trigonometry, the
Stereographic Projection of the Sphere is given ; and among
many other things, two chapters, that will be found very use-
ful to those who are desirous of studying practical Astronomy^
The fourth book includes a brief treatise on the theory and
practice of Navigation, which, as we observed in the begin-
ning of this chapter, is one of the branches of practical sci-
ence that depends almost wholly on trigonometry. The volume
concludes with a number of extremely useful tables, among
which are a table of logarithms from 1 to 10,000 to five
places of decimals ; a table of natural sines to every degree
and minute of the quadrant; and a table of logarithmical
sines and tangents also to every degree and minute of the
quadrant.

Besides the works avowedly on Trigonometry, we have an
excellent work entitled " A Practical Introduction to Spherics
and Nautical Astronomy, being an attempt to simplify those
useful sciences." By P. Kelly, LL.D : and another " On
the Doctrine of the Sphere," in six books, by the Rev.
George Walker, F. R. S., containing Preliminary properties
of the cone the general doctrine of the Sphere and spheric
triangles Projection, orthographic and stereographic, and a
treatise on Spheric Trigonometry. This work is, we believe,
very scarce, and has at no time been appreciated according to
its merit.

It is usual to have tables of Logarithms in a separate
volume; of these there are many entitled to commendation;
and, in speaking of them, we cannot follow a more respect-
able authority than that of Mr. Bonny castle, who, in his
trigonometry, says, " among the most accurate and conveni-
ent of whiph, for common use, may be reckoned the edition
of Vlacq's small volume of tables, printed at Lyons in 1670



490 MATHEMATICS.

and another work of this kind, printed at the same place, in
1 760 ; but more particularly the edition of Sherwin's Mathe-
matical Tables, in 8vo. 1 742, as revised by Gardiner ; also
Hutton's Mathematical Tables, in 8vo. first printed in 1785;
the Tables of Vega, 2 vols. Bvo. printed at Leipzig , in 1797;
and the first edition of the Tables Portatives de Logarithms
of Callct, in small 3vo. printed at Paris, 1783 ;* all of which
are adapted to the sexagesimal division of the circle, used by
Valcq and most of the later compilers.

" Besides these, several other tables, of a different kind,
have been lately published by the French ; in w hich the quadrant
is divided, according to their new system of measures, into
100 degrees, the degree into 100 minutes, and the minute
into 100 seconds ; the principal of which are the second edition
of the Tables Portatives of Callet, beautifully printed in ste-
reotype, at Paris, by Didot, Bvo. 1 795, with great additions
and improvements ; the Trigonometrical Tables of Borda,
in 4to. an. ix, revised and enriched with various new precepts
and formulae by Delambre ; and the tables lately published at
Berlin, by Hobert and Ideler, which are also adapted to the
decimal division f the circle, and are highly praised for their
accuracy by the French computers.

" Among the various tables, however, of the sexagenary
kind, none have been more esteemed for their usefulness and
accuracy than those of Gardiner, printed in 4to. at London,
in 1742; which contain the logarithms of all numbers from 1
to 102100, and the logarithmic sines and tangents for every
ten seconds of the quadrant, to 7 places of decimals, with
several other necessary tables ; a new edition of which work
was also printed at Avignon, in France, in 1770, under the
care of Pezenas, who added to it the sines and tangents of

TbU neat portable work, which is now become extremely scarce, con-
taios all the tables in Gardiner's 4to. volume, hereafter mentioned, with
several additions and improvements ; and is, by far, the most useful and
convenient performance of the kind that faai yet been offered to the
public.



TRIGONOMETRY. . 491

every single second, for the first 4 degrees, and a small table
of hyperbolic logarithms, taken from Simpson's Fluxions.

*' But of all the trigonometrical tables hitherto published,
the most extensive and best adapted for obtaining accurate
results, in many delicate astronomical and geodetical observa-
tions, are those of Taylor, printed in large 4to. at London,
1792; which contain the logarithms of the first common
numbers from 1 to 1260, to eight places of decimals; the
logarithms of all numbers from 1 to 101000, to 7 places;
and the logarithmic sines and tangents of every second in the
quadrant, to 7 places ; as also a preface, and various precepts
for the explanation and use of the tables, which, from the
author's dying before the last sheet of bis work was printed off,
were supplied by Dr. Maskelyne, the late astronomer royal.

" It may here likewise be observed, that besides the common
tables hitherto mentioned, which contain the logarithms of
numbers in their usual order, others, of a different kind,
have been constructed, for the more readily finding the num-
ber corresponding to any given logarithm ; of which the prin-
cipal one, of any considerable extent, is the Antilogarithmic
Canon of Dodson, published at London, in 1 742 ; which
contains the numbers corresponding to every logarithm, from
1 to 100000, to eleven places of figures, with their differences
and proportional parts ; and, though little used at present, is
a performance of great labour and merit."-*

With respect to the new division of the circle, by the French,
referred to in the preceding page, and the advantages of which
have been so highly applauded by many authors, we may ob-
serve, that, though it would unquestionably facilitate calculation,
yet it would render useless all existing trigonometrical and
astronomical works, as well as the most valuable mathemati-
cal instruments, which have been constructed and divided by

Dr. Wallis informs us, in the 2d vol. of his mathematical works, that
an antilogarithmic canon was began by Harriot, the algebraist (who died in
1621), and finished by Warner, the editor of his works, about the year
I640j but which was lost for want of encouragement to print it.



492 MATHEMATICS.

die most eminent artists. The best instruments in all tlie
observatories of Europe, have been made by British artists,
as Ramsden, Troughton, &c., and they are all graduated
according to the sexagesimal division of the circle, therefore,
supposing the change to take place, recommended by the
French, it is manifest that all observations, made with these
instruments, must be reduced to the centesimal division of the
circle, before they can be used in calculation. All the lati-
tudes and longitudes of places on the globe must likewise be
changed, which change would render the different works on
Geography, in a great measure, useless ; or otherwise, those
in the habit of making trigonometrical calculations, must be
perpetually turning the old divisions of the circle into the new,
or the new into the old. The same may be said of all the
logarithmical tables of sines, tangents, &c., so that the change
from the sexagesimal division to the centesimal is a matter of
doubtful advantage, when balanced with the difficulties that
must necessarily attend it.



CHAP. XXXII.



MATHEMATICS,

Continued.



Conic Sections Method of obtaining the several sections History of
the Science. Writers upon it Apollonius Pappus Wallis Hamilton
Robertson De la Hire Simson Bosco vicb Newtop ^Vince Jack
Trevigar Steel, &c. Fluxions rules for finding the fluxions of quan-
tities application of fluxions writers on fluxions Newton Rowe
Vince Simpson Maclanrin. Doctrine of Chances Annuities, Insur-
ance, &c.



do NIC Sections are such curve lines as are produced by
the mutual intersection of a plane, and the surface of a solid
cone. The nature and properties of these figures were the
subject of an extensive branch of the ancient geometry, and
formed a speculation well adapted to the genius of the Greeks.
In modern times the conic geometry is intimately connected
with every part of the higher mathematics and natural philo-
sophy. A knowledge of the great discoveries of the last
century, cannot be attained without a familiar acquaintance
with the figures and properties of the conic sections.
These sections are derived from the different ways in
which the solid cone is cut, by a plane passing through it ;



494 MATHEMATICS.

and they are, a triangle, a circle^ an ellipse, a parabola, and an
hyperbola. The last three of these, are pecuHarly called
Conic Sections, and the invesligation of their nature and pro-
perties, is generally denoted by the term " Conies." The
mode of obtaining these sections is as follows :

(].) If the cutting plane pass through the vertex of the
cone, and any part of the base, the section will be a triangle.

(2.) If the plane cut the cone parallel to the circular bjise,
the section will be a circle, provided the cone be a right
one.

(3.) The section is a parabola when the cone is cut by a
plane parallel to the side ; or, when the cutting plane and
the side of the cone make equal angles with the base.

(4.) The section is an ellipse, when the cone is cut oblique-
ly through both sides, or when the plane is inclined to the
..base in a less angle than the side of the cone is.



Online LibraryWilliam ShepherdSystematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) → online text (page 40 of 44)