University of California • Berkeley 
 
 Purchased from 
 CHABOT AND DIECKMANN 
 MEMORIAL LIBRARY FUND 
 
THE 
 
 Young Ladies and Gentlemen s 
 
 AUXILIARY, 
 
 IN 
 
 TAKING HEIGHTS AND DISTANCES, 
 
 CONTAINING THE 
 
 Use of the Small Pocket Case 
 
 OF 
 
 MATHEMATICAL INSTRUMENTS, 
 
 ILLUSTRATED BT 
 
 Practical Geometry^ 
 
 THE 
 
 Knowledge of which is essential for the Student in 
 Geography and Astronomy, as well as in Mensura- 
 tionj Perspective and Architecture. 
 
 PART I. 
 
 SECOND EDITION. 
 
 BY T. DRUMMOND. 
 
 Bacon^ KinncbrooJCj and Co, I*rinters^ Cockcy^lane^ Norwich, 
 
CONTENTS. 
 
 Greek Alphabet. Page, 
 
 Greek, Latin, and English Numerals. 
 StcT. 1. Geometrical Deiininitions, with Remarks ,. 1 
 
 2. Definitions relative to Angles ♦ 6 
 
 3. Erecting Perpendiculars, Dividing Lines, &c. 
 
 with the Compasses « . . 8 
 
 4. Inscribing Figures in Circles 13 
 
 5. Of tlie Protractor, Plain Scale, and Diago- 
 
 nal S^ale •••.,,••..> 18 
 
 Geometrical Theorems relative to Angles , , 25 
 
 Theorems relative to Circles 26 
 
 ^. Of Quadrangles or Quadrilateral Figures .. 
 Theorems relative to Four-sided Figures , . 29 
 
 7. Of Right Lined Triangles 
 
 Theorems relatiVe to Triangles 3@ 
 
 S. Trigonometrical Problems for laying down 
 
 tlie Sides of Right Angled Triangles .... 32 
 9. Of the Quadrant, &c 35 
 
 10. Trigonometrical Problems for Ascertaining 
 
 the Angles and Measuring the Sides or Legs 
 of Right Angled Triangles 37 
 
 11. Oblique Angled Triangles ., •*... 40 
 
 12. Of Solids ^... 4^ 
 
 13. Planimetry, or the Measnrement ojT Plane 
 
 Surfaces ....* ,.•#.... 4$ 
 
 14. Stereometry or the Mensuration of Solids . . 
 The Five Platonic Bodies 49 
 
 15. Altimetry and Longimetry, or the Mensura- 
 
 tion of Ileighths, Depths, and Distances. . 51 
 
 16. JViisceiiauea relatiye to Geography, &c, . . . 66 
 
The Pupil is particularly requested to make the fol- 
 lowing corrections : 
 
 Page 12. For by elliptical, read but elliptical. 
 
 21 . For 90 equal parts, read 9 equal parts. 
 
 S3. For figure 73, read figure 76. 
 
 36. For figure 74, read figure 77. 
 
PREFACE 
 
 TO THE FIRST EDITION. 
 
 THE present age has been remarhahle for its aliempts 
 to faciliiatc the progress of youth. The elevated genim 
 tf Mrs, Barhauld has eondeseended to assist the itifant 
 7nind in the ueqitirement of rational ideas : — Dr, Aihift 
 has converted the thoughtless school-bo?/ into an admirer 
 of the ivorhs of nature, and a more general diffusion of 
 knoziiedge has been happili/ effected bi/avarietT/ of suc^ 
 ccssful attempts to smooth the rugged paths of literature. 
 A happy union of the Useful and agreeable seldom fails 
 to amuse and interest us ; and though industry meets with 
 no diffcidties zchich are insurmountable, yet the rapidity 
 of her progress must depend on the nature and number 
 of the obstacles she has to encounter. She scales the 
 mountain s top, or penetrates into the bowels of the earth 
 — encompasses sea and land, or measures the arc of 
 heaven — exit nds her calcidations from world to world^ 
 and from one sy stein to another, with more or less fatigue 
 in proportion to the advantages she enjoys. There is an 
 inactivity in the human mind analogous to the inertness 
 of matter ; every action must be produced by the impulse 
 of some motive^ and the art of the teacher principally 
 
PREFACE. 
 
 ronsisis in being ahk to present such motives to the mindj 
 as may produce the desired effects. 
 
 The mathematics are geneially supposed to abound 
 with uninteresting and tedious theories^ not inducible to 
 practice without long application mid great fatigue, — 
 But as the traxeller^ whose sofe object is recreation, no 
 sooner arrives at the gate of a building famed for being 
 the repository/ of many curiom productions of art and 
 nature, than he is anxious to obicun admission ; in like 
 vwnner the human mind, become intimately acquainted 
 with t fie introductory parts of science, feels an impulse to 
 explore its less obvious magnificence and beauty. 
 
 The system of education, at this time generally adopt- 
 ed, expands the mind a^id enlarges its sphere of action. 
 The broad basis of general information is by some re* 
 garded as dangerous, lest it should hinder the attainment 
 of that degree of perfection, in any particular science^ 
 zchich a more limited application might produce, — By 
 general Imowledge a man becomes a more extensively 
 useful member of society ; and whenever genius finds 
 opportunity for any particular application, the favorite 
 passion is pursued with an ardor ?iot less fervent, and 
 probably affords pleasures not less delicious than Ms, who 
 never steps beyond the periphery of a circle of a given 
 radiusy or who never thinks but according to the standard 
 of some classical autliority. 
 
 Young people, moreover, are often placed in situations 
 Qiherse to theirnatur at genius, which excite disgust when 
 
PREFACE* 
 
 ihey arrite iit i/fars ofmaturUy; a contracted edttccdmn 
 unfits them for any other pursuit whereas a more liberal 
 plan of instruction would enable them to profit by any 
 change of circumstances^ and extend the sphere of their 
 utility. 
 
 The universal attention that is paid to Geography at 
 this time; and the elements y at least ^ of Astronomy being 
 considered as essential in<tny liberal plan of female educa- 
 tion^ led to a suggestion^ that a familiar explanation of the 
 use of the smaller cases of instruments^ with an account 
 of some easy methods of taking heights and distances ^ 
 might perhaps contribute to the more ready diffusion of 
 hiowledge on such subjects. 
 
 If this little tract should contribute to furnish young 
 persons with a rational amusement^ repel an antipathy 
 to the mathejnaticSjCind excite a desire of further investi- 
 gation of a subject which instructs us to explain the 
 various plwenomena of the heavenly host^ and a variety 
 of authors will be sought for with avi4ity^ and afford 
 fimple satisfaction* 
 
PREFACE 
 
 TO THE SECOND EDITION. 
 
 E VER Ypar€7it as zcell as exerij teacher is azoare that 
 wformation is more effectuaUi/ conveyed to the minds of 
 young persons' by familiar conversation than I)?/ elaborate 
 writings on any subject. 
 
 The success of the tutor ^ in his mode of exciting at'^ 
 tention and interest in his pupils^ depends on his varying 
 the language of communication according to the quicks 
 7iess of comprehension which each pupil manifests — but 
 the occasional mode of arguing^ the similes^ SsC. to con- 
 xey ideas most successfully^ the interrogatory process to 
 develope the hidden fund of juvenile Ifiozdedge already 
 acquired^ would appear ungraccfidly in prints because 
 the reader cannot be minutely informed of the 72umerous 
 xarying discriminations requisite between cases of i^eal 
 ignorance and of diffidc7it i^eserve. 
 
 Ask the scholar whether xchat has been said be well 
 understood — the answer will probably be an affirmcdive. 
 Every teacher knows that yes is a spontaneous ef^ctofa 
 desire not to appear inattentive, and that 7nore interroga- 
 tion and moi^e circumlocidio7i are necessary to asceriain 
 the degree of infor7nation actually acquired^ but it is ////- 
 possible to lay before the public the inquisitorial p rocesscs 
 in the court of tuition. 
 
PREFACE. 
 
 ISeries of questions may he ]udldouslij arranged and 
 usefully emyAoyed^ many remarks and queslions zchieh 
 ihe tulor^ acting ae cording to circumstances^ cannot pre- 
 meditcde.^ will however he found necessary, 
 
 TJie strength of inemory zchich some children possess 
 will enahle them to repeat rerhalini the language of the 
 author they have studied^ it may therefore be expedient 
 to vary the language of the questions and to accept an- 
 swers in the youths' own phraseology in order to cievelopc 
 the latent worhings of intellect. 
 
 Grammar is frequently learned hy rote — Geography 
 is learned hy rote — History is learned hy rotc^ hut it is 
 ohvious to any one who cd tends to the consequences, that 
 the words of the hook adopted are repeated with 7io more 
 exercise of the energies of the mind than a parrot is 
 competent to display, provided the lessons were so short 
 as to be incessantly repeated. 
 
 Geography and the use of the Globes are almost uni- 
 xersally taught, but unless a prober foundation be 
 laid — unless ihe geometriccd parts of the science he under- 
 stood— 7/oung ladies and young gentlemen will speak 
 of parcdlels of Icditude and meridian li?jes withoid any 
 precise idea annexed to either of them. 
 
 The follozoing gradation of instruction is intended not 
 only tofcuiUtate an acqucdntance with Prcutical Geome- 
 try, but to introduce to the minds of young persons the 
 knowledge of several things requisite to be understood in 
 an early stage of their intellectual improvement. 
 
PREFACE. 
 
 The former edition conledned the use of the Sccforj 
 Logarilhms^ S^c, hut as tkh first volume comprises what 
 is most important for beginners m Mensuration, Geo^ 
 graphs/ J and the Draumg Maps of the easiest construe-' 
 Hon — the second volume will contain the latter part of 
 the first edition^ with additions relative to Practicat 
 Astronomy. 
 
 As the knowledge of the derivation of terms gives coT' 
 rect ideas of their meaning, it is an effectual method of 
 impressing them on the memory/. 
 
 The most obvious etifmology is adopted in ihe Auxili^ 
 ary^ and the same method will be observed in all the 
 future publications relative io the sciences. When the 
 Latin and Greek terms are nearly similar both are in" 
 seited; where the derivation can be easily traced to only 
 one source, that alone is noticed. 
 
 A knowledge of the Greek Alphabet is a necessary 
 acquisition io those who wish to become familiur with 
 ihe Ccelestial Globe, as (he Stars are designated by the 
 Greek letters to express their different magnitudes — thus, 
 tf, ihefrst letter in ihe Greek Alphabet, is assigned io the 
 largest star in each constellation — 0, the second letter, to 
 ihe next in size, S^c. and when the number of stars ex* 
 ceeds the number of letters^ ihe Italic letters are intrcduced* 
 A Greek Alphabet is therefore prefixed to this small 
 publication. Young persons in general will learn the 
 Greek Alphabet as an amusement, and en hour'^s appli- 
 cation will render them familiar with ihe characters. 
 
 Norwich, 1814, 
 
GREEK ALPHABET. 
 
 f^ere arr Twentg-fmr L^ers m GrceL 
 
 A, et^ 
 
 Al^ha, 
 
 ii* 
 
 B, ./3, C, 
 
 O^ta, 
 
 h. 
 
 r* yy f. 
 
 C^amm^j 
 
 ^^ 
 
 A, A 
 
 Delta^ 
 
 d. 
 
 r, e, 
 
 Epsilon^ 
 
 ^. 
 
 2,{. 
 
 Zeta, 
 
 z. 
 
 H, rf. 
 
 Eta, 
 
 ^. 
 
 e, 9, ^, 
 
 Theta, 
 
 th. 
 
 I. /, 
 
 Iota, 
 
 1. 
 
 K, X, 
 
 Kappa, 
 
 i. 
 
 A, A, 
 
 JLambd(£, 
 
 J. 
 
 M, ^, 
 
 Mu, 
 
 in. 
 
 N, V, 
 
 Nu, 
 
 n. 
 
 H,e, 
 
 Xi, 
 
 X. 
 
 O, 0, 
 
 Omicron^ 
 
 d. 
 
 n, 'T, 'sr^ 
 
 Pi, 
 
 P- 
 
 P| p, 
 
 Rho, 
 
 t. 
 
 :S, <r, filial (T, f. 
 
 Sigma, 
 
 4. 
 
 T, T, 7, 
 
 Tau, 
 
 t. 
 
 T, t/, 
 
 fjpsilon. 
 
 ij. 
 
 *^j ^ 
 
 Phi, 
 
 |jb. 
 
 ^^ X^ 
 
 Chr, 
 
 ^-h. 
 
 % 4, 
 
 Psi, 
 
 ps. 
 
 Q, ^s 
 
 Om^ga, 
 
 3* 
 
NUMERALS. 
 
 Greek, Latin. English^ 
 
 ?K, 4. uiuis, one. 
 
 cvo^ duo, two. 
 
 T^zl^^ tres, , three* 
 
 'r'c(Ta'cLoi<:, quntuor, four. 
 
 ^ini, quinque, five. 
 
 er, * • sex, six. 
 
 ^-TTTct, septem. seven. 
 
 c/.T»5 octo, eight. 
 
 ivuict, noveni, nine. 
 
 J iaci, decern, ten. 
 
 k:'cf^i)ta, ,. undecim, eleven, 
 
 Sd<^iKu.y diiodecim, twelve. 
 
 rpictKctiJ^iKct, . . . tredecim, thirteen. 
 
 Ti(T(TipAK3Li'S'iKct, quatuordecim, ,,. fourteen. 
 
 "TTivr^KoiiJ^Ziia.,, ,4 quindecira, fifteen. 
 
 iKKniJ^ix^d, sexdecim, sixteen. 
 
 cTTety.cLicf'iKct, . . . septenid^cini, . , , . seventeen* 
 
 oKTc-JKctiJ^iKa, . . . octodecini, eighteen. 
 
 liyzAKitiJ^i^ct, ,, . liovemdecinij .... nineteen. 
 
 uaoiTt, viginti, twenty. 
 
 UKo<n siV, unus et viginti, .. twenty-one 
 
 kl^S'-'ju.:]y.ovra.: Q')-J^oi}Ky/Tcti iym'-'iKOi'Tcc, IkatqV. 
 
9inj:ilm^, $c* 
 
 MATHEMATICAL* INSTRUMENTS. 
 
 A SMALL Case of Instruments contains a pair of 
 compasses, one of whose points may be taken off and 
 a drawing pen for ink or a crayon case for a lead 
 pencil substituted in its place. 
 
 A semicircular or parallelogram protractor, a black 
 lead pencil, a scale of 4^ or 6 inches, according to 
 the size of the case, are also included. 
 
 The protractor may be obtained separately when 
 the case has not been furnished with it. 
 
 * Ma9>j/xct Science. Tid. Protractor and Plain Scale. 
 
 SECTION L 
 
 Geometrical * Definitions, with Remarks. 
 
 A Point is the minutest expressible part of a line, 
 and may be conceived more minute than can be de» 
 scribed by a pencil or pen, — Fig. 1. 
 
 * r« the earth and fAirpv measure, 
 c 
 
s 
 
 »'i'» i y «> ff« ^ il«ga»^^l»SWi»\F6 ' V *^--tS^'^:^^BfW»gi'«^^ 
 
 «■ -*i* JPV-* *SjgJ5^Bf»S»,V«»j»<./^ 
 
 A Line — consists of^a number of close] j conti- 
 guous points. — Fig. 2. 
 
 A STRAIGHT OR RIGHT LiNE — lics cvcn between 
 its extreme points ; and only one right line can be 
 drawn between any two points. 
 
 A Cur VE Line — deviates higher or lower between 
 its extreme points. — Fig. 3. 
 
 A Serpentine Line; a continuation of alter- 
 liate curves. — Fig, 4. 
 
 Parallel Lines, whether straight or curved, 
 are equally distant from each other in every part, 
 and consequently can never meet. — Hg, 5, 
 
 Workmen suspend freely a plumb * or plummei 
 hy a string, which is called a plamh line ; when it 
 ceases to vibrate, it settles in a right line, inclining 
 neither to the right nor the left side; — the line is then 
 called perpendicular and hangs in a direction towards 
 the centre of tlie earth. — Fis;, 6. 
 
 *o* 
 
 * Plumbum^ lead — the metal generally used for the 
 weight. The plumb lin^ U CQinmonly fastened to the top 
 of a board grooved to an open space, in the middle of which 
 the weight hangs when the line is perpendicular. 
 
 The joiner'^s square is sometimes furnished with aline and 
 weight, as in the former instrument. 
 
 ^, Circle. — If with any opening of the compasses 
 one point be fixed on papcr^ &c. whUst the other de- 
 scribes a line around it in every part equally distant 
 from the fixed point, that curved line is called the 
 
^Circumference or the fperipherT/; the space con- 
 tained witliin the circumference is called tlie cifrle^ 
 and the fixed point the centre of the circle. — Fig- 7. 
 
 * Circum^ around, fero^ to bear, f H^p/, around, ^spcy, 
 to bear. 
 
 Diameter,! — Aline drawn from one point of the 
 circumference of a circle through its centre to the 
 opposite point, dividing the circle into two equal 
 parts. — Fig. 8. 
 
 X S'ltt through, and ^STpof measure. 
 
 Radius.^ — A line from the centre of a circle io 
 the circumference — consequently the radius is half 
 the diameter. — Fig, 9. 
 
 The plural of radius is rudii, 
 
 § VctCJ^o^. Radius^ a ray of light, or spoke of a wheel, 
 i. e. a right line from the centre. 
 
 Chord. II — A line passing from one part of the 
 circumference of a circle to another, but not passing 
 through the centre. — Fig. 10. 
 
 Jl %op</^»f, chorda, a string of a bow. 
 
 Anc.5 — ^Any part of the circumference of a 
 circle. 
 
 % Arcus^ a bow. 
 
 c 2 
 
Segment.* — The space contained between tlie 
 the chord and the arc ol a circle. 
 
 * Seco^ to cut off. 
 
 Every circle, whatever its diaracter, is Universally 
 considered as divisible into 360 parts, called degrees, 
 and marked thus — 360^. 
 
 Each degree contains 60 minutes, marked tlius,' — 
 each minute contains 60 seconds, marked tlius. " Tlie 
 divisions are carried farther in Astronomical calcula- 
 tions, 60"Hhirds, one second, &c. 
 
 S60 is a preferable number, because it may be di- 
 vided into halves, quarters, and half-quarters — 180'', 
 90% 45^ 
 
 Let a circle be described and the diameter W E 
 be drawn — open the compasses and find the point N 
 in the circumference, at an equal distance from W 
 and E — then draw a right line from that point, pass- 
 ing through the centre to the opposite point S. 
 
 You will then have two diameters, and the circle 
 will be divided into four parts, containing 90° each; 
 let the quarters of the circle be in like manner divi- 
 ded and tw o more dianieters drawn — the number of 
 radii will then be 8; — draw lines from each point to 
 the third beyond it, and either darken one half of 
 each point with Indian ink or apply different colours 
 to the halves of each. — Fig* 11. 
 
Let each space between tlie points be divided as 
 before and t lie re will be 16 — divide again and there 
 l^ill be 32 points. — Fig\ 12. 
 
 Figure 12 is a representation of a compass card used 
 by mariners and surveyors, and frequently placed be- 
 tween the feet of the claw-footed stands of the ter- 
 restrial and celestial globes; — a magnetic needle, turn- 
 ing freely on a point, naturally places itself in the 
 direction North and South — whence it is easy to de- 
 termine the bearings, or on \yhat point of the com- 
 pass each of the surrounding places lies. 
 
 Because the Sun rises in the East, passes io the 
 South of us, and sets in the West — hold the figure so 
 that E may be towards the place of the Sun's ap- 
 pearance in the morning, and E will indicate the 
 East, W the West, N the North, and S the South, 
 both with respect to the earth and the heavens. 
 
 Cardinal* Points. — The four points, East, 
 West, North, South. 
 
 Rhumbs. — The spaces between the four princi- 
 pal points, are subdivided into 8 points, which arc 
 called llhumhsj so that the compass contains 32 
 points, each rhumb containing 11^ degrees. These 
 are described on the compass card, and also on 
 the wooden horizon belonging to the globes. From 
 the North towards the East these points are named 
 
 * Cardinalls^ principal. 
 
NorUi and by East— North North-east— North- 
 east by North — North-east — North-east and by 
 East — East North-cast — East and by Nortli, East, 
 &c. 
 
 In the same manner the points are described from 
 the East towards the South. 
 
 Concentric Circj.es. — Two or more circles 
 having the same centre. — Fig, 13. 
 
 Let two concentric circles be drawn, and draw 
 two diameters so as to divide the circles into four 
 quarters — divide the upper semicircle into 18 divi- 
 sions, calling each division 10", numbering them 
 from to 180 ; — if the half of each division be marked 
 off, each space will represent 5° — and if the circle 
 be sufficiently large, each single degree may be de- 
 scribed with a fine pen. — Fig. 14* 
 
 SECTION IL 
 
 The following Definitions relative to Angles 
 will now be easilij understood. 
 
 An Angle is the space contained between two 
 right lines, drawn gradually nearer to each other till 
 they met in a point, as A C B. — Fig, 15. 
 
 Angular Poi.^t. — The point where i\\e two 
 lines meet; in describing the angle we read A C B, 
 placing C, the letter at the point, in the middle. 
 
A Right Angle is formed %vlien the space be- 
 tween the two lines is equal to 90**, i. c. a quarter of 
 a circle* — Fig. 16. 
 
 In a right angle the lines forming the angle are 
 perpendicnlar to each other. 
 
 Not more than four right angles can be described 
 about the same point, because each right angle is one 
 quarter of a circle. 
 
 Acute* Angle. — Aji angle containing fewer 
 than 90o.—Fig. 17. 
 
 Obtuse f Angle. — An angle containing more 
 than 90\—Fig, 18. 
 
 The complement of an angle is what it want« 
 ofOO^ 
 
 The supplement of an angle is what it wants 
 of 180^ 
 
 *■ Acutus^ sharp. 
 
 f ObtusuSy not pointed, i.e. less pointed than a Riglvt 
 
 The Measure of an Angle Described. 
 
 Lay a thread from the centre C, fig. 14, to SO'', in 
 the semicircle, and the thread line will form with the 
 line C D an acute angle of 30% i. e. B C f). ^0\ 
 
 Apply a thread from centre C, ilg\ 11, to 120% 
 arid ttie thread line wdll form aii obliise angle with 
 tiie line C D. 
 
8 
 
 SECTION III. 
 
 Of Erecting Perpendiculars, Dividing Lines, 
 ^c. with the Compasses. 
 
 To ^Bisect any given Line., as A B, into Two Equal 
 Parts.— Fig. 19. 
 
 With an opening of the com passes to more thau 
 half the given line — place one foot on B, one extre- 
 mity of the given line, and describe an arc — then on 
 A describe another arc, + intersecting the former in 
 two pointsj and through the intersections draw the 
 line C D. 
 
 ^BiSy in twain. Secoy to cut. + Inter^ between. 
 
 To Erect a Perpendicular from a given Poitit A in the 
 Line A i?.— Fig. 20. 
 
 Describe a circle of any radius, so that the point 
 B may be in the circumference — from D the other 
 points in A B cut by the circle draw a diameter, and 
 from the termination of the diameter draw the line to 
 EB. 
 
 To Erect or Let Fall a Perpendicular on any Point 
 in a given liight Line. — Fig. 21. 
 
 Let C be the given point in the line A B — at 
 equal distances from C mark D and E, and witli any 
 opening of the compasses greater than C E — place 
 
9 
 
 one foot on E and describe a small arc as F — place 
 the same foot on D and intersect the arc F by the arc 
 G — ^then from the point of intersection draw a line to 
 C, which will be tlie perpendicular required. 
 
 To Erect a Perpendicular on the Point B in the Line 
 A ij.— Fig. 22. 
 With any opening of the compasses set oflf Ba on 
 A B — fix one foot on a as a centre and describe the 
 arc C B — with the same radius from B intersect the 
 arc C B in d. On d as a centre describe F B — set 
 oft' two extents from B in B F, and from the second 
 draw a right line to B. 
 
 To Erect a Perpendicular on the Point B in A JB.— 
 Fig. 23. 
 With any opening place one foot of the compasses 
 on B and describe an arc as F a — with one foot on F 
 intersect the arc at C — on C describe the arc d f — on 
 f describe C b, and draw a line through the point of 
 intersection to the angular point. 
 
 To Draw a Li?ie Parallel to a Line given. — Fig. 24. 
 
 With the compasses opened to the distance of tlie 
 proposed parallel, taking points in the given line as 
 centres, describe two arcs and draw the new line so as 
 • to touch these arcs without intersecting]^ them. 
 
10 
 
 To Divide (tn Angle into Two Equal Parts. — -Fig. 23. 
 
 On the angular point as a centre describe an arc 
 as a b — then on a and b, with the same radius, de- 
 scribe arcs intersecting each other, and draw a line 
 from the angular point to the point of intersection. 
 
 2o Divide a Line A B into any number of Equal Paris, 
 Fig. ^6. 
 
 Draw B C any angle and A D parallel to B C— 
 
 lay off on B C and A D, the number of equal parts, 
 
 for instance /?re, into which A B is to be divided, and 
 
 join the opposite points by straiglit lines passing 
 
 through A B. 
 
 To Describe a Ilelix'^ or Spiral Line '\- A B. — Fig. 27. 
 
 Draw A B and divide half of it into as many parts 
 as there are to be revolutions of the proposed spiral, for 
 instance fotir^ thus, c d e f , then divide e f in g — and 
 on g, as a centre, describe the semicircles above the 
 line — and on f, as a centre, describe tlie semicircles 
 below the line* 
 
 * $A/|, a screw, firxs/py., a line like a screw. 
 
 To Describe an Oval ton a given Line A B. — Fig. 28. 
 Divide A B into three equal parts — on C and D, 
 as centres, describe circles intersecting each other, 
 and with the exteiit A D on the points of intersection 
 as centres describe tlie arc above and the arc below 
 to complete the (igur?. 
 
 t OvutJK Jin cgi;. I'lic resemblance consists in one 
 diiimeter beiui; slsortcr than tlie other. 
 
11 
 
 To Describe an Oval from Three Circles, — Fi<^. 2d. 
 
 Draw A B and divide it into four equal parts — on 
 the tliree points D C E describe tlirce circles — draw 
 a diameter, F G totlie middle circle at right angles 
 with A B. On G5 with an extent from G to the re- 
 motest part of the peripheries of the outer circles, de- 
 scribe the arc II I, and with the same extent on F 
 describe the arc K L. 
 
 To Describe an Oval, the Short Diameter being three- 
 fourths of the Long Diameter, — Fig. 30. 
 
 Draw A B and C D at riglit angles, and divide the 
 short diameter into four equal parts. With an open- 
 ing of the compasses equal to three-fourths of the 
 short diameter make the points b and d centres in de- 
 scribing the arcs e C f and a D c — and with an open- 
 ing equal to half the long diameter make the point 
 P the centre of two arcs also, and the figure will be 
 completed. 
 
 To Delineate a Circle Txith a Thread and Pencil. 
 Fig. 31. 
 
 Stick a pin upright — take a thread and tie its ends 
 together to form a loop, the length of the proposed 
 radius. Ij^t one end of the loop be around the pin 
 and the other extended by the insertion of a black 
 lead pencil — move the pencil round the pin as far 
 distant as the loop will permit — the place of the pin 
 
12 
 
 will be the centre, and tbe line described by the 
 pencil will be the circumference or periphery of the 
 circle. 
 
 An Ellipsis is a regular oval, possessing properties 
 not found in the ovals above-mentioned. 
 
 Instruments as the Elliptical Compasses, the Geo- 
 metric Pen, &c. are employed in drawing Ellipses, 
 but as they are not likely to be in the hands of those 
 for whose use this part of the Auxiliary is intended, 
 it may be sufficient to notice one method of deli- 
 neation. 
 
 To Delineate an Ellipse ^"^ having the Long and Shwt 
 Diameter s;iven. 
 
 Draw the two diameters A B and C D — with an 
 extent equal to half the long diameter, place one foot of 
 the compasses at either end of the short diameter and 
 intersect the long diameter at F f, which points of 
 intersection will be the fzco foci A 
 
 Erect a pin at each focus and form a loop around 
 them equal in length to A B — then by the insertion of 
 a pencil, as in the delineation of a circle, describe the 
 Periphery of the Ellipse. 
 
 The Planets do not move in Circular by Elliptical 
 Orbits — the Sun being in one of the Foci. 
 
 * sAAs/x^y, to be deOcient— one diameter being shorter 
 than the^ther 
 
 + ^fejf'fir, to burn. 
 
13 
 
 SECTION IV. 
 
 Of Inscrihing Figwes in Circles, 
 
 A figure described by three equal sides is called a 
 Trigon. Four equal sides at right angles consti- 
 tute a Square. 
 
 To Inscribe an Equilateral^ Triangle t or Trigon % 
 in a Circle, — Fig. 32. 
 
 Draw the radius A C — with the extent of radius 
 set one foot on A and intersect the circumference on 
 each side, as B and D — draw the line B D, which 
 will be the extent of one side of the proposed 
 triangle 
 
 * Equus, equal. Laius^ a side. 
 + Tres^ three. JnguluSy an angle. 
 % TpzKTy three, ysovioti an angle. 
 
 To Inscribe a Square in a Circle, — Fig. 33. 
 
 Draw the diameters A B and C D at right angles, 
 and draw the chords C A, A D, B D, and B C. 
 
 Polygons, § 
 
 § IToXi;^, many, yaviet, an angle. 
 
 All figures having more than four sides are deno- 
 minated Polygons — if the sides of the figure arc 
 
14 
 
 equal it is called a Regular Folygoii-7-if unequal , an 
 Irregular Polygon. 
 
 To Inscribe a Pent.hgon ^'" in a Circle, — Fig. Si, 
 Draw two diameters at right angles — divide the 
 radius C D into two equal parts, on E with the ex- 
 tent E A describe the arc A F — and on A describe 
 F G — then draw the line G A, which will be the 
 measure of one side of a pentagon. 
 
 * nij'Tss five, yxvtdi^ an angle. 
 
 To Inscribe a Hexagon \ in a Circle. — Fig. 35. 
 
 Set off the radius of the circle six times on the cir- 
 cumference and draw the chords to the six arcs. 
 
 i E^, six. ycdvict^ an angle. 
 
 To hiscribe a Heptagon % in a Circle, — Fig. 3(5. 
 
 Draw the radius A C — with that extent set one 
 foot of the compasses on A and describe the arc 
 b C b — draw the riglit line b b, and half b b set off 
 seven times on the circumference will mark the limits 
 of the sides of the Heptagon very nearly. 
 
 X "E-TTTct^ Sepfein^ seven, yuvia^ an angle. 
 
 To Inscribe an Octagon % in a Circle, — Fig. 37. 
 
 Half an arc of the side of the square will be the arc 
 of one side of the ©ctagon. 
 
 ^ Oktco^ Odo^ eight, yccvtcty an angle. 
 
15 
 
 To Inscribe an Enncagon * or Nonagon f in a Circle. 
 Fig. 38. 
 
 Set off the extent of one tliird of the arc of a side 
 of tlie triangle nine times on the circumference. 
 Or ^vith more geometrical accuracy thus — 
 Draw a radius A B — with the extent of radius and 
 one foot on B describe the arc C A D — draw C E D 
 at the extremities of the arc, and extend it so as to 
 make E D F equal to radius — upon the point F de- 
 scribe the arc E G and draw the line A G — on E de- 
 scribe F G, to whicli from D draw'D H, the side of 
 the polygon proposed — ncarlij. 
 
 * ivvict* f Novem^ nine. y!>jvia, an angle. 
 
 To Inscribe a Decagon % in a Circle, — Fig. 39. 
 
 Half the arc of one side of a pentagon will be the 
 arc of one side of a decagon, 
 
 % ^iKdi^ Deccniy ten. yc':vii;.j an angle. 
 
 To Inscribe an Endecagon % in a Circle, — Fig. 40. 
 
 Draw the radius A B, bisect it in C — with an 
 opening of the compasses equal to half the railii-s, 
 upon A and C as centres describe the arcs GDI and 
 A D — witli the distance I D upon 1 describe the arc 
 D O and draw the line C O, which will be the extent 
 
 § '^S'iKct^ Uiidecem^ eleven, ycovici, an angle. 
 
16 
 
 of one side of an endecagon sufficiently exact for 
 practice. 
 
 To Inscribe a Dodecagon * in a Circle. — Fig. 41. 
 
 Half the arc of one side of an hexagon will be the 
 arc of one side of the dodecagon. 
 
 * j^coS'iA.dL, Duodecirrij twelve, ycovicty an angle. 
 
 To Inscribe a Polygon in a Circle, — Fig. 42. 
 
 Divide the diameter A B into equal par(s, corres- 
 ponding to the number of sides in the proposed 
 Polygon. 
 
 Erect a perpendicular from the centre, and extend 
 it to C, three-fourths of radius beyond the circum- 
 ference; — draw the line C D through the second di- 
 vision of the diameter, and the chord D A will be 
 the side of a heptagon, the polygon required in this 
 instance. 
 To Inscribe liegular Figures in a Circle, — Fig. 43. 
 
 Draw the two diameters and with radius, on the 
 extremities of each, describe arcs and number the 
 divisions on the circimiference from SO to S60. 
 
 SO will be the side of a dodecagon — 60 the side of 
 a hexagon — 90 the side of a square — 120 the side of a 
 trigon. 
 
 Draw aline from the centre through the intersec- 
 tions of two arcs and the cliord A B will be the side 
 of an octagon. 
 
17 
 
 To Ascertain the Centre of a Circle. — Fig. 44, 
 
 Draw any chord A B — bisect it m ith a perpendi- 
 cular equal to the extent of the circle-— bisect the per- 
 pendicular, and each part will be a radius, and con- 
 sequendy the point where the two radii meet is the 
 centre. 
 
 To Describe the Circiimfercjice of a Circle through 
 three given Points ^ not in a Straight Line. — Fig. 45. 
 
 Draw two lines to connect the middle point with 
 the others; — on the middle of those lines erect per- 
 pendiculars and extend them till they meet. — On 
 the point of intersection with its distance from one 
 of the given points describe the circle. 
 
 By this method the centre of acircle may be ascer- 
 tained ; or if only an arc be given, the centre may 
 be found and the circumference of a circle com- 
 pleted — or a circle may be circumscribed about a 
 given triangle. 
 
 The number of triangles into which a polygon may 
 be divided and its number of sides are equal. 
 
 The angles of any polygon, regular or irregular, 
 added together amount to twice as many right angles^ 
 excepting four, as the figure has sides. 
 
 Regular polygons may either be inscribed in or 
 described about a circle. 
 
18 
 
 SECTION V. 
 
 Of the Protractor. 
 
 The Protractor is usually of a semicircular fornij 
 graduated into 180« and is numbered botli ways for 
 the conveniency of laying down or measuring angles 
 in any direction. 
 
 The plain edge is the diameter, and a fine mark^ at 
 the middle of it, is the centre of a circle. 
 
 The Protractor is sometimes in the form of a 
 ruler, to which the divisions are transferred from the 
 semicircle — the outline of two figures will serve to 
 shew the correspondence between them. — Fig. 46. 
 
 To Lay Down an Angle B of ib\—Y\g, 47. 
 
 Draw the line A B, place the edge of the Protrac- 
 tor exactly upon it, and the central mark on the point 
 B, then make a dot at 45^^ C, and a line drawn from 
 C to B will describe the angle required. 
 
 To Measure an Angle. 
 
 Place the centre of the protractor on an angular 
 point, and the plain edge exactly on the line A B, 
 the number of degrees cut by the line C B will be the 
 measure of an angle B 45"*. 
 
 To Erect a Perpendicular to a given Ldnc. 
 Let the angle laid down be 90", and one liiie will 
 be perpendicular to the other. 
 
19 
 
 Polygons. 
 
 A« every circle is supposed to be divided into 360«, 
 that number divided by the number of sides in the 
 polygon required, has for its quotient the number of 
 angles at the centre; and if the angles at the centre 
 be subtracted from 180°, the remainder will be the 
 angle at the circumference. 
 
 A TABLE 
 
 Of Angles ai the Centres and Circumferences of Re gu- 
 
 lur Poli^'gons. 
 
 Sides. 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 Names, 
 
 Trigon . . 
 Square • . • 
 Pentagon • 
 Hexagon • 
 Heptagon • 
 Octagon . 
 Nonagon • 
 Decagon 
 Endecagon . 
 Dodecagon . 
 
 Aiigles at ihe 
 Centre, 
 
 .. 120° 
 
 • . 90° 
 
 . • 72 
 
 ..60 ... .• 
 
 . . 45 
 
 ..40 
 
 ..36 
 
 ..30 . . . . , 
 
 Angles at the 
 Circumference. 
 
 , . 60 
 
 .. 90 
 .. 108 
 . 120 
 
 128, 34 f 
 
 135 
 
 HO 
 
 144 
 
 147,16/r 
 150 
 
 To Inscribe in a Circle a Polj/gon of any number of 
 Sides. 
 
 Apply the diameter of the protractor to the dia- 
 meter of the circle, and let their centres coincide ; — 
 E 2 
 
20 
 
 mark the degree given for the angle at the centre of 
 the proposed polygon and draw a radius, then a 
 chord of an arc equal to the given angle will be one 
 side of the polygon required. 
 
 The Plain Scale. 
 
 There are usually six or seven scales of different 
 dimensions on the side called the Plain Scale, 
 
 Each scale is divided into equal spaces, which 
 bear a certain proportion to an inch as one-third, one- 
 fourth, &c. The number at the beginning of each 
 scale, as SO, 25, SO, &c. indicates into how many 
 parts an inch is divided. 
 
 A space equal to one division on the scale is marked 
 with 10 subdivisions — the fifth, for the sake of dis- 
 tinction, is a longer stroke than the others. 
 
 If one in the scale be designed to represent one 
 foot, yard, mile, league, &c. the compasses extended 
 from 1 to 3 in tlie smaller divisions w ill express one 
 and three-tenths — from 2 to the 5th subdivision will 
 express two and five tenths, &c. which extent set olf 
 on a line will represent one and three tenths of a mile, 
 yard, &c. 
 
 T^esize of the plan of any drawing will vary ac- 
 conii - 'J the scale employed. 
 
 If 1 be called 10, the subdivisions will be so many 
 units — if I be reckoned 100, the subdivisions will be 
 tens — if 1 be called a 1000, the subdivisions will be 
 hundreds. 
 
21 
 
 On the best scales the upper line is markcfl into 
 12 subdivisions, of which the Sdj Gth, and 9thj par- 
 ticularly the 6th, are longest. 
 
 3b set off 530^ or 53^ or 5 arid three tenths. 
 Place one foot of the compasses at 5 on the large 
 divisions, and extend the other foot to o amongst the 
 subdivisions. 
 
 To express 4 feet 8 inches zcith a scale zchose 2/pper line 
 is divided into 12 parts • 
 
 * Place one foot of the compasses on 4 amongst the 
 large divisions and extend the other foot to 8 amongst 
 the subdivisions — and mark oW that extent on any 
 proposed line. 
 
 The Line of Chords, 
 
 The large divisions on the uppermost scale end 
 sooner than the rest, to admit r line of chords, marked 
 Cho or C and numbered to 90^. 
 
 The line of chords is used in preference to the pro- 
 tractor, for laying off or measuring angles. 
 
 To Construct a Line of Chords. — Fig. 48. 
 Divide the arc of one quarter of a circle into 90 
 equal parts, containing 10° each — draw the chord, 
 and with one foot of the compasses on A as a centre, 
 transfer the divisions of the arc A B to the chord 
 AB — annex the corresponding figures to the right 
 
22 
 
 line A B, and it Mill be a line of chords. A chord 
 of 60 is equal to radius^ i. e. A 60' is equal to 
 AC. 
 
 To Laj/ Dozoi an A)2^Ie of 25° at the Point A in the 
 Line FA. — Fig. 49. 
 
 Set oiie foot of the compasses at C, the beginning of 
 ^he line of chords, and extend the other to 60 — then 
 on A as a centre describe an arc F G — take the ex- 
 tent from C on the line of chords to So^, and set it off 
 on the arc FG — draw the line G A, and the angle 
 F AG will equal 2j'\ 
 
 To Measure an Angle hy the Line of Chords. 
 
 With a chord of 60"^ place one foot of the com- 
 passes on the angular point and describe an arc — then 
 lake the extent of the arc contained between the two 
 lines forming the angle, and apply one foot to the 
 begining of the line of chords on the scale, and the 
 other will extend to the degree which will denomi- 
 nate the measure of the angle — thusthe angle FAG 
 given will be Sj*". 
 
 Describe a Circle with Radius of GO"" taken from the 
 Line of Chords^ and Divide 360 hi/ the number of 
 Sides in the proposed Figure. 
 
 90 on the line of Chords will give one side of a Square. 
 
 72 • • • • a Pentagon. 
 
 60 t • • t • • I ff • 1 1 • • t • t an Hexagon. 
 
23 
 
 45 on the lineofCIiords will iciveonesideofanOctacron 
 
 40 an Enneagon or Nonagon. ' 
 
 36.. •• •••••••••a Decagon. 
 
 30.,.,. •••••• • a Dodecagon. 
 
 The Heptagon and the Endecagon not being di- 
 visible bj their respective numbers^ may be found 
 by other methods. ^ 
 
 If the line of chords on the scale be too large for 
 the proposed circle, describe tfie line of chords on the 
 diameter extended, and draw a line from the measured 
 degree to the centre of the circle. 
 
 TifE DiAGOyjL ScJtLE. 
 
 On the oilier side of the >ca]e is (' hie of inches ac- 
 cording to its length, divided like rulers in general 
 into inches and tenths of an inch. 
 
 Also a line of equal parts, which shews the foot to 
 be divided into an 100 equal parts. The six-inch 
 scales extend to 50, and thiis 3 inches are 25 parts ot 
 the said foot. 
 
 The scale diaganallj/ divided is used for the same 
 purpose as the plain scales already described — but 
 whilst the small divisions on the plain scales express 
 the te?iths of one of the large divisions, the diagonal 
 scale is adapted to a greater degree of exactness; it 
 serves to express the hundredth part of one of the 
 large divisions, for which reason it is called a Cen- 
 teshimal* Scale. 
 
 * Centum, an hundred. 
 
At one end of tlic scale is a part equal to oneof tlic 
 large divisions, divided at the top and the bottom 
 into 10 parts, which, by diagonal lines drawn from 
 the 10th beiovv to the 9th above, subdivide the said 
 10 parts into a hundred less proportional parts, by 
 which means not only the 10th of a number may be 
 expressed but the 10th of that 10th. 
 
 Thus for 245. — The second large division ex- 
 presses 200 — t]ic fourth diagonal expresses four tenths 
 or 40, and the fifth parallel expresses 5. 
 
 Tliere an* two diagonal subdwisions^ one at eacli end 
 of the scale — the one is exactly half the length of the 
 other. 
 
 The larger divisions are numbered 1, 2, 3, 4, &c. 
 according to the length of the scale. The half-sized 
 divisions are numbered at the bottom. 
 
 Ten [parallel lines, from one end to the other, are 
 numbered 2, 4, 6, 8, at the ends, and tlie diagonals 
 are also nmnbered 2, 4, 6, 8 on the top, by which the 
 lines are readily distinguished. 
 
 Set off 284. 
 Place one foot of the compasses where the unit 4 
 intersects the diagonal of the tens 8, and extend the 
 other loot to the hundrrds expressed by the large di- 
 visions — th' s place one foot where the 4th parallel 
 meets the 8Ui diuironal, and extend the other foot to 2 
 amongst the large divisions. 
 
25 
 
 Set off 567. 
 
 Place one foot of the compasses where the 7th pa- 
 rallel meets the 6th diagonalj and extend the other 
 foot to 5 amonfi^st the large divisions. 
 
 Observe, — If the part diagonally divided b»called 
 1, then the subdivisions are of the same description 
 as those on the plain scales, i. e. tenths of one. 
 
 If the part diagonally divided be called 10, theu 
 the large divisions express tens, the number of the 
 diagonal expresses the 10th parts of ten, and the 
 number of the parallels shews the ten minute parts. 
 
 If the part diagonally divided be called 100, thea 
 the numbers ou the large divisions express hun- 
 dreds — the number in the diagonals expresses tens^ 
 and the number on the parallel notes units — thus, 
 284, 567, 632^ to bG set off on a given line. 
 
 Geometric.1l Theorems Rel.itive to Angles, 
 
 Theorem I. — Fig^ 50. 
 
 A right line A B forms with another line C D two 
 angles, the measures of which addod together arc 
 equal to two riglit angles — thus, if AB I) be 64°^ 
 ABC will be 116'', the number of degrees in the 
 semicircl® being 180^ 
 
 Theorem 2. — Fig. 51. 
 
 The oppdsite angles made by two right lines in- 
 tersecting each other are equal, as a aind b, c and d. 
 
 V 
 
26 
 
 Theorem 3. — Fig, 59. 
 
 A riglit line intersectii^g two parallel lines makes 
 the opposite angles at each intersection equal. 
 
 2'jiEOREi^s Relative to Circles. 
 Theore?7i 1. — Fig. 53, 
 
 The angle at the centre of a circle is double the 
 angle at the circumference ^vlicn both of them stand 
 on the same arc, L e. A B C is double the angle 
 ADC. 
 
 The measure of an angle at the C€?7lre is the arc 
 contained betMcen the lines forming the angle — but 
 the angle at the circumferctice may be proved by 
 measurement to be only half the arc at the centre. 
 
 Theorem 2. — Fig. 54. 
 
 An angle in a semicircle* is a right angle at tlie 
 Ciircnmference. 
 
 The arc ADC which measures the angle ABC i« 
 a semicircle, half which is £)0°. 
 
 * Semis^ half. 
 
 Theorem 3. — Fig. 55. 
 
 An angle made in a segment greater thaa a semi- 
 circle is acute y as B A F. 
 
 Theorem 4. — Fig. b6. 
 
 An angle in a segment less than a semicircle is 
 obtuse^ as may be proved by measuring the angles. 
 
%% 
 
 SECTION VI. 
 
 Of Quadrangles^ or Quadrilatei^atj' Figures. 
 
 Tetragon J is a term applicable to every figure 
 baying four angles, 
 
 * QuatuoVj four. Angulus^ an angle. 
 
 \ Quatuor^ four Latus^ a side, 
 
 X TiT^ct, four. ycoviAy an angle. 
 
 Tgrp^fc is for rirr^pet, and rzmpct for rz(r(JzpdL^ 
 
 To Describe a Square. — Fig. 57. 
 
 Draw the line A B, one side of the proposed 
 square — at one end of the line erect a perpendicular 
 with the protractor, or the line of chords, or any of 
 the preceding directions. — Make the perpendicular 
 equal to the side A B, and with the extent of A B on 
 A and C describe arcs, and the point of intersection 
 will determine the boundaries of the figure. 
 
 To Describe a Parallelogram* — Fig. bS. 
 
 A right line figure, which like a square contains 
 four right angles, but whose opposite sides only arc 
 equal. 
 
 Draw A B one of the long sides — erect a perpen- 
 dicular B C equal to one of the short sides, and with 
 the extent of A B on C describe the arc at D, and with 
 the extent of B C on A intersect it by another arc, 
 and draw A D and C D to complete the figure. 
 
 F 2 
 
28 
 
 To Describe a Rhombus* — Fig. 59. 
 
 A Rhombus is formed by four equal lines and con- 
 tains four angles, two of which are acute and two 
 obtuse. 
 
 Draw the side A B and lay down the angle at A. 
 Suppose 36° — draw A C equal to A B, and with in- 
 tersections as in describing the square draw B D 
 and C D. 
 
 To Describe a Rhomboides. — Fig. 60. - 
 
 A Rhomboides differs from a parallelogram as a 
 rhombus differs from a square — draw one side and lay 
 down the angle A — suppose 45^*, and proceed as in 
 the parallelogram io complete the figure. 
 
 To Describe a Trapezium, — Fig. 61, 
 
 A Trapezium is a figure bounded by four unequal 
 sides and containing four unequal angles. 
 
 Take the given lines from the plain or the diagonal 
 scale and connect them according io the given 
 angles. 
 
 To Describe a Trapezoid.-^Fig. 62. 
 
 A Trapezoid differs only from a trapezium by hav-" 
 ingtwo of its four sides parallel. 
 
 Irregular Poli/gons, — Fig. 63, 
 
 All unequal right-line figures under more than four 
 sides are called irregular polygons. 
 
29 
 
 The directions for describing the trapezoid nnd the 
 irregular polygons are the same as given above for 
 the trapezium. 
 
 Theorems Relative to Four-sided Figures, 
 Theorem 1. — Fig. 64. 
 
 The four inward angles gf a quadrangle or qua- 
 dilateral taken tooether are equal to four right 
 angles. 
 
 Theorem 2, — Fig. 65. 
 Parallelograms on the same base and between the 
 same parallels are equal, i. e. A B C D is equal te 
 B E F C. 
 
 SECTION VIL 
 Cff Jlig^f^t Lined Triafigles. 
 
 A figure bounded by three right lines is called a 
 Triangle. 
 
 A Right Angled Triangle is distinguished by one 
 of its angles being an angle of 90"". — Fig. 66. 
 
 An Obtuse Angled Triangle has one obtuse angle. — 
 Fig.Ql. 
 
 An Acute Angled Triangle has its three angles 
 acute. — Fig, 68. 
 
 An Equilateral Triangle has its three sides equal.— 
 Fig. 69. ' 
 
30 
 
 Ayi Isosceles^ Triangle has only two sides equal. — • 
 Fig. 70. 
 
 * I^of, equal, XkaKo^, a leg, 
 
 A Scalene + Triangle has its three sides unequal. — 
 
 Fig.ll. 
 
 f ^KctKvwo^j oblique and unequal. 
 
 In every triang-le tJie longest side and the greatest 
 angle are opposite to eacli other. 
 
 To Describe a Triangle, — Fig. 72. 
 
 Draw one side as A B of the extent required — on 
 A, with the extent of another line A C, describe an arc 
 at C — auvi with the extent of the third lineB C inter- 
 sect the arc at C — then draw lines to complete the 
 figure. 
 
 Theorems Relative to Tri^ngle&. 
 
 Theorem 1. — Fig. 73, 
 
 In every triangle tlie sum of all the angles is equal 
 to 180^5 i. e. two right angles. 
 
 Therefore — If any two angles in a triangle be 
 known, llieir sum taken from 180° gives the remain- 
 ing angle. 
 
 If owe angle be a right angle ^ the other two angles 
 added together equal the other right angle. 
 
31 
 
 If dne angle in a triangle be a right or an ohtusc 
 angle, the other Izoo angles aie acute. 
 
 If m a right angled triangle one of the acute angles 
 be known, subtract it from OO"", and the remainder is 
 the other acute angle. 
 
 Theorem 2, — Fig. 74. 
 
 If one side of a right angled tria?\gle be produced, 
 the outward angk will be equal to the two inward 
 opposite angles. 
 
 Obs€7've, — In every right angled triangle the side 
 opposite the greatest angle is called tlie Hypotenuse,* 
 the upriglit side the perpendicular, and the other 
 side the Base, t 
 
 * vTTo^ under, rzivc-:, to extend, 
 t B^flo;, the lowest place. 
 
 Theorein 3. — Fi 
 
 'to* 
 
 In a right angled triangle the square % of the hypo- 
 tenuse or longest side is equal to the square of the 
 base and the square of the perpendicular added 
 together. 
 
 % Tlie square of a numher is found hy mnltipljlng it 
 by the same number, thus — 3 multiplied by 3 is 9, the 
 squaie of 3. 
 
 An inspection of the figure illustrates this proper- 
 ty of the right angled triangle, the knowledge of 
 
32 
 
 wbich is of great use in most branches of tlie Ma- 
 tliematics.* 
 
 * Pythagoras, a Celebrated Grecian Philosopher, is re. 
 corded to have been the first who made this discovery. 
 
 " It is suid, that he was so elated after making the dis- 
 roxery^ that he made an oiferlng of a heratomb to the 
 Gods; but the sacrifice was undoubtedly of small- oxen, 
 made with wax, as the philosopher was ever an enemy to 
 shedding the blood of all animals. His System of tiie 
 Universe, in which he placed the ^Sun in the centre, 
 and all the Planets moving in elliptirat orbits round it, 
 was deemed chimerical and improbable, till tiie deep en- 
 quiries and tlie philosophy of the 16th century proved it 
 by the most accurate calcidations to be true and incon- 
 testable.'' — Lemprierc''s Diet, 
 
 SECTION VIII. 
 
 Trigonometrical Problems for Laying Down 
 the Sides of Right Angled Triangles, 
 
 Problem 1. — Fig. 76. 
 
 The Base and Perpendicular given to find 
 the Hypotenuse. 
 
 Given. Pcquired, 
 
 A B Base 20. A C Hypotenuse. 
 
 B C Perpendicular 15. 
 Draw a line and set off from a scale of equal parts 
 A B 20 — erect a perpendicular on the point C — opeiL 
 the compasses to 15 on the same scale as before — ap- 
 ply that extent to the perpendicular from the point 
 Ji and intersect B C in C — then draw the line A C — 
 
35 
 
 the extent of which applied to the scale will be 25, 
 the length of A C, the hypotenuse. 
 
 Problem S.— Fig. 75. 
 
 The Base and the Hypotenuse given to find 
 the Perpendicular. 
 
 Given. Bequired, 
 
 A B Base SO. B C Perpendicular. 
 
 A C Hypotenuse 50. 
 
 Draw the base and set off SO from the scale of equal 
 parts — erect a perpendicular atB — take from the same 
 scale the length of the hypotenuse 50, and with one 
 foot on A extend the other to intersect the perpendi- 
 cular in C — from that point of intersection draw the 
 line C A, by which the triangle will be completed— 
 apply the extent of B C to the scale, and it will be 
 40, the perpendicular required. 
 
 If the Hypotenuse and Perpendicular were given, 
 and the Base required, the operation would be the 
 same, substituting Perpendicular instead of Base — 
 thus, 
 
 Given. Required, 
 
 B C Perpendicular. A B Base. 
 
 A C Hypotenuse. 
 
 In the above trigonometrical problems for deter- 
 mining the sides of right-angled plain triangles, two 
 sides were given and only one side remained to be 
 sough!. 
 
34 
 
 In those cases wherein the perpendicular .^ i. e. th« 
 height of the object, is not known, and the distance 
 of the observer, viz. the base of tlje triangle, does not 
 admit of the application of any mcasnre, and no line 
 can be extended from the top of the object to the 
 place of the observer, in order to ascertain tlie hypo* 
 tcnuse ; it is by this time obvious to the pupil that 
 neither height nor distance can be determined ; 
 for unless two of the lines could be previously known, 
 the iltird cannot be determined by the foregoing 
 rules. 
 
 The invention of opticaly mathematical, and philo- 
 sophical instruments, has abundantly contributed to 
 increase the rational amusements of mankind; the 
 principles on which they are constructed are not so 
 uninteresting as they may at first appear; every 
 theorem has its correspondent practical operation ; 
 every problem admits of a solution applicable to 
 some of the purposes of life. The benefits resulting 
 to society from such studies are innumerable. — ^Each 
 individual amongst us is at least a practical geome- 
 trician, and the mind naturally acquires much more 
 knowledge of the theory of geometry than is usoally 
 apprehended. Neitlicr extraordinary labour, nor 
 much time is requisite for gaining a competent know- 
 ledge of those scientific truths which are of use ia 
 the common concerns of life. And we may venture 
 io assert, that an attention to mathematical rules^ is an 
 
35 
 
 inexhaustible source of recreation, conduces to the 
 promotion of our interest, and enables the mind most 
 clearly to discern the excellent order in the system of 
 the universe, seeing that " God lias ordered all things 
 in measure, number, and weight." 
 
 The ingenious contrivances of mankind to in- 
 crease the faculty of vision, by means of telescopes ;^ 
 and with quadrants, theodolites,* &:c. to ascertain the 
 heights and distances of accessible or inaccessible 
 objects, are to be numbered amongst tlie evidenceis, 
 which prove that the power God has bestowed on 
 man was appropriated to him for the investigation of 
 truth, and the observance of the harmony of the celes- 
 tial system. 
 
 * 020//*/, to see. oJ^o^, distance. / 
 
 SECTION IX. 
 
 Quadrant, Sgc. 
 
 Amongst the variety of ingenious and useful in- 
 ventions which have been adopted for taking the 
 angle of observation, it will be sufficient for the illus- 
 tration of the following problems if two only be 
 mentioned. 
 
 The instruments called Quadrants are variously 
 
 contrived, and furnished with different apparatus, 
 
 according to the uses for wliich they are designed. 
 
 Every quadrant is the fourth part of a circle, having 
 
 Q 2 
 
36 
 
 the arc divided into 90°, and subdivided more or less 
 according to tlie size of the instrument — everj 10th 
 degree being drawn longer than the rest. 
 
 If the juvenile mechanic wishes to supply himself 
 with a quadrant, let him take a smooth flat board, and 
 havingdescribed upon it an exact quarter of a circle, 
 and graduated the arc as in fig. 74, place a pin at 
 the centre and from it suspend a thread and plum- 
 met — the heavier the plummet the more steadily it 
 will hang, and the observation may be made more 
 correct; — on the side B C let two thin plates of brass, 
 with very small holes, exactly in a riglit line, be 
 placed parallel to the side of the quadrant. 
 
 Another simple instrument is called a Quadrat, or 
 Geometric Square — it consists of four rules joined 
 together at right angles ; at A a thread and plummet 
 is suspended, and occasionally a moveable index at- 
 tached ; the sides B E and D E are divided into 50 
 or 100, and if the instrument be large, into 1000 
 equal parts — two sights are fixed on the side A D, 
 similar to those above-mentioned. 
 
 The quadrant invented by Mr. Gunter, whose 
 name it bears, has also a quadrat delineated upon 
 it, and is not only useful in taking the angle of vision 
 but serves *for a variety of astronomical problems, 
 which will be described in the second Vol. 
 
 Ganter's Qnadrant being an instrument not liable 
 to accident and of a moderate price, is most likely to 
 be in the hands of yoiing persons. 
 
m 
 
 To Take mi Angle of Altitude with the Quadrant. 
 
 Hold the quadrant in the direction B C, so that 
 the top of the object may be viewed through both 
 sights at the same time — and the plumb line will cut 
 the arc of the quadrant in the number of degrees and 
 parts of a degree equal to the angle of altitude, i. e. 
 the arc of the quadrant will be equal to the angle 
 ACT. 
 
 When the object is higher than the eye of the 
 spectator, the angle taken is the angle of alti- 
 tude — when the object is lower than the eye of the 
 observer, it is called the angle of depression. 
 
 To take an Angle of Depression. 
 
 Apply the eye to the centre and observe the angle 
 on the other side of the plumb line. 
 
 SECTION X. 
 
 Trigonometrical Problems for Ascertaining 
 the Angles and Measuring the Sides or Legs 
 of Right Angled Triangles, 
 
 Problem 1. 
 The Base and Perpendicular ^ i. e. the two Legs of 
 
38 
 
 a Right Angled Triangle given tojind the Acute Angles 
 and the Hi/potenuse. 
 
 Given. Required. 
 
 A B 45. Angle A. 
 
 B C 80. Angle C. 
 
 AC. 
 From either the plain or diagonal scale, draw the 
 base A B 45 and the perpendicular B C 80, and com- 
 plete the construction of the triangle by drawing the 
 hypotenuse A C. 
 
 Measure one of the acute angles with tlie arc of 
 60 from the line of chords, the other will be the 
 complement of the first — and apply the extent A C 
 to the same scale used for the base and perpen- 
 dicular. 
 A7isxi).—kng\G A 60- SO'. 
 
 A ngl e C 29*^ SO '. Hypotenuse 92 nearly. 
 
 Problem 2. 
 
 The Hypotenuse and one Leg of a Right Angled 
 Triangle given to find the Angles and the a her Leg. 
 Given. Required. 
 
 Perpendicular B C 75. Angle A. 
 
 Hypotenuse AC ISO. Angle C. 
 
 Base A B. 
 Draw a line, on which the base line A B is to be 
 set off. 
 
 On B erect the perpendicular B C 75, reckoned 
 on a selected scale; take 130 from the same scale, 
 
39 
 
 and on C as a centre, with a radius of ISO, describe 
 an are intersecting the line drawn for the base at 
 A — complete the triangle by drawing the line A C. 
 then measure one of the acute angles and the base. 
 Amw. — Angle A S5<^ 15 ^ Angle C 54° 45'. 
 Base A B 106 and 2 tenths. 
 
 Problem 3, 
 
 The Angles arid one Leg of a Right Angled Tri- 
 angle given to complete the figure^ and find the other 
 Leg and the Hypotenuse, 
 
 Given, Required. 
 
 Angle A 65° 50'. Perpendicular B C. 
 
 Base A B 400. Hypotenuse A C. 
 
 Draw A B equal to 400 measured on a diagonal 
 
 scale — erect a perpendicular at B — at the point A 
 
 form an angle Qb"" SO , and draw A C till it extends to 
 
 the perpendicular. 
 
 The complement of angle A will be angle C — B C 
 and A C applied to the same diagonal scale which 
 was used for the base, will give 
 Perpendicular B C 877. 
 Hypotenuse A C 964|. 
 
 Problem 4. 
 
 The Angles and Hypotenuse given to find the Legs 
 of a Right Angled Triangle, 
 Given. Required. 
 
 Angle A 5P. Base A B. 
 
 Hypotenuse A C 474. Perpendicular B C, 
 
40 
 
 Draw the line A B of an indeterminate length ; at 
 A form an angle 51° — with the line AC, from a 
 diagonal scale, make A C represent 474 ; let fall a 
 perpendicular from C, meeting the base in the 
 point B. 
 
 Angle C is known by its being the complement of 
 angle A — let A B and B C be measured by the same 
 diagonal scale used in setting off xA. C, and the answer 
 will be 
 
 Base A B 298. 
 
 Perpendicular B C 368. 
 
 SECTION XI. 
 Oblique Angled Triangles. 
 
 Problem 1. 
 
 Two Angles of an Oblique Angled Triangle j and 
 the side opposite to one of the Angles given to find the 
 other side. 
 
 Given, Required, 
 
 Angle C 6Q\ A C 
 
 Angh^ A bOo, B C 
 
 A B 450. 
 Add together the two given angles and sub- 
 tract their sum ?rom 180« — the remainder will be 
 angle B. 
 
41 
 
 Draw A B 450 — ^lay down angle A 50« and angle 
 B 70° — A C and B C meeting in a point complete 
 the figure. 
 
 Answ.—A C 488 and 3 tenths. B C 398. 
 
 Problem 2. 
 
 Two Sides of an Oblique Angled Triangle and an 
 Angle opposite to one of them, given to find the other 
 Angles and the third Side. 
 
 Given. Required. 
 
 A B 525. Angle A. 
 
 A C 475. Angle C. 
 
 B 5K B. C. 
 
 Draw the line A B 525 — at B make the angle 51''— 
 take the extent of A C as a radius, and on A as a 
 centre intersect B C in C — then draw A C to complete 
 the figure. 
 Answ. — Angle A 69''|, i. e. 45'* 
 Angle C 59^1, i.e. 15'. 
 B C 575 and 6 tenths. 
 
 Problem S. 
 
 Two Sides and the Angle formed hy them I given to find 
 the third side and the remaining Angles. 
 
 Given. Required. 
 
 A C 352. Angle A. 
 
 B C 266. Angle B. 
 
 C 13\ A B. 
 
 H 
 
42 
 
 Draw A C and B C, making- a» angle 73^ at C, 
 and draw A B to complete the figure. 
 
 Apply the extent of A B to the scale — Answ, 374. 
 
 Measure one angle either A or B — add the angle 
 found to the angle giveuy and the third angle being 
 the supplement of the sum of the two angles, will be 
 known by subtracting that sum from 180", 
 
 ArTswers, — Angle A 42« 45', Angle B 61" la 
 A B 374. 
 
 SECTION XII. 
 
 Of Solids. 
 Thesquare,* the rhombus^f the parallelogram.:}: the 
 rhoraboides,^ the triangle, &c. already noticed, relate 
 only to the superficies or plane surfaces, bounded by 
 the lines of the respective figures, and are therefore 
 said to have only two dimensions — length and 
 hreadlh. 
 
 * Quadra^ Quatra, from quatuor. 
 
 + PifxCc'y, to deviate. The angle* of the figure deviate 
 from right angles. 
 
 The term Rhumh^ page 5, h from the same origia 
 Pi^.Cft). 
 
 When a ship saih in any other direction than to 
 (tne of the four cardiaal point<t, her course is described 
 by a rhumb line. If the coarse were continued in an- 
 oblique direction, it would not dejfcribe a circle, hut a 
 figure mx)re resembling a spiral. 
 
 X ricip^tAAMAof, mutual. ypctfjifjAi^ a liae» 
 
 4 PoiaCo^^ rhumb. g/cTof, form. 
 
43 
 
 Solids — arebotlies which havo three dimensions — 
 kngtfi^ hr^adlh^ and thickness. 
 
 A Cube* is a solid contain^xl bj six square 
 sides.— JP/g-. 78, 
 
 ACuBOlDESf OrPARALLELOPIPEDiUN^isaSGlid 
 
 having six qa*adrangular surfaces^ every opposite pair 
 equal and parallel. — Fig. 79. 
 
 t KvCoc, a cube, gz/o', form. 
 + TstpcthMiKot. TOcTcf, feet 
 
 A Prism § is a solid whose ends are similar, equal, 
 and parallel, and its sides parallelograms. — Fig. 80. 
 
 § npta^a^ from ^^lu^ to cut with a saw. 
 
 A prism of solid glass in an instrument employed 
 for separating the rays of solar 1| light. 
 
 II Sol^ the Sun. 
 
 If the young philosopher 5 form a prism with 
 three pieces ofglas^, and securing the ends with clay, 
 wood or metal, iill the prism with water, he may be 
 gratified by an exhibition of a series of beautiful 
 colours and be prepared to admire the more splen- 
 did elfects of a glass prism accurately ground and 
 polished. 
 
 f ^/Aor, a lover. 'Zo^tct^ wisdom* 
 
 H 2 
 
The terra prism is, however, applied by geometri- 
 cians to any figures whose ends are similar, parallel, 
 and equal, and whose sides are parallelograms — thus 
 the ends of prisms may resemble the triangle, the 
 square, the pentagon, the hexagon, &c. 
 
 A Pyramid * is a solidhavingany of the different 
 figures for its base — triangular sides meeting Ia a^ 
 point at the top. — Fig. 81. 
 
 * Tivp, fire, because flame ascends to a point. 
 
 A CoNEt is a solid having a circular base — the 
 figure gradually decreasing towards the vertex or 
 top, and ends in a point. — Fig. 82. 
 
 A Cylinder J is a solid having circular ends and 
 being circular in its whole extent. — Fig. 83. 
 
 % KuKivJ^c^j to roll. 
 
 A Sphere § or Globe is a solid whose surface is 
 imiform, and therefore every part equally distant 
 from a point within called (he centre of the sphere, — 
 rig. 84. 
 
 § Sphera. 
 
 A Segment is a part cut off the top of any solid 
 parallel to the base. — Fig. 85, 
 
45 
 
 A Frustum* is the part remaining after thcsen^- 
 ment is cut off. — Fig. 80. 
 
 * Frustum^ a fragment. 
 
 A ZoNE+ is that part of a sphere which is be- 
 tween two parallels. — Fig, 87. 
 
 t Zsov\u a belt. 
 
 The term Zo7ie is particularly applied Xofrce divi- 
 sions of the earth — the spaces consisting of 23f de- 
 grees between the Equator \ on tlie Gloi^e and tlie 
 Tropic % of Cancer J and between the Equator 
 and the Tropic of Capricorn might be called two 
 Zones, but as they are contiguous to each other and 
 comprise the whole of that space over some part of 
 which the Sun in its daily apparent progress becomes 
 perpendicular, it is usual to consider them as 07te 
 Zone^ denominated the Torrid. |j 
 
 J Equus^ equal. The circle divides the globe into 
 two equal parts, the Northern and Southern hemis- 
 pheres. 
 
 § Tp/TTftj, to turn — because the Sun appears to turn or 
 change its course back again when arrived at either of 
 the circles denominated the Tropics, which are two 
 small circles parallel to the Equator — the former pass- 
 ing through the beginning of tlie sign Cancer — the latter 
 through the beginning of tlie sign Capricorn. 
 
 [! Torreo^ to scorch. 
 
46 
 
 The spaces between tlie parallel of the Tropic of 
 Ccmcer and the Arctic Circle^ and between the Tropic 
 of Capricorn and i\\Q Antarctic Circle^ are denominated 
 the North and South Temperate Zones; and the 
 spaces cut off by the parallel of the Arctic Circle and 
 by the parallel of the Antarctic Circle are termed 
 Frigid Zones. 
 
 ApKTo;, a bear. 
 
 Aj/t/j against or opposite A^kto^* 
 Temperantia^ moderate heat. 
 J^iyio^i frigidus^ cokl. 
 
 SECTION XIIL 
 
 Planimetry,^ or tim Measurement of Plane 
 Surfates, 
 
 Although this publication is not intended as a sub- 
 stitute for a treatise on mensuration, it may be use- 
 ful to supply i\ic pupil with the simple rules for the 
 measurement of plane surfaces. 
 
 * Planus^ plain. piirpiK, to measure. 
 
 Numerous opportunities offer for rendering the ap- 
 plication of the rules a source of amusement — as the 
 superficial content or area of a sheet of paper, a tabk, 
 or carpet, a side zeal! of a room, an entire room, a 
 garden, Sfc, may be ascertained by them. 
 
47 
 
 To Measure a Qiiadr angular Figure whose opposite 
 Sides are Equal. 
 
 Rule. — Multiply the length by the breadth. 
 
 Thus a square, each of whose sides is 5 iaches, or 
 feet, &c. will be 25 inches, feety &c. in area* or 
 content. 
 
 A Parallelogram^ 18 in length and 6 in breadth, 
 will Ije found 108 in area. 
 
 * AreUj a space contained between lines. 
 
 The Rhorabus and tlie Rhoniboides, whose angles 
 are not right angles, must have their breadth ascer- 
 tained by erecting or letting fall a perpendicular. 
 
 To Measure am/ Right Angled Triangle. 
 
 Rule. — Multiply the base by half the perpendi- 
 cular, or the perpendicular by half the base, i. e. mul- 
 tiply one leg forming the right angle by half the 
 other. 
 
 To Measure any Triangle, 
 
 Rule. — Ascertain the perpendicular height and 
 multiply the base by half the perpendicular licight, 
 or the perpendicular height l)y half the base — tht^ 
 product will be the area. 
 
 To Measure a Trapezium *\ 
 Rule. — Divide the figure into triangles and com- 
 pute each separately, and tlien add the sums together. 
 
 f Tf^Te^/sr, a small table. 
 
48 
 
 To Measure a Ti^apezoid. * 
 HuLE. — Add the parallel sides and multiply half 
 tlieir sum by the distance between them. 
 
 To Measure an Irregular Polijgon* 
 
 Rule. — Resolve the ligure into squares, paral- 
 lelograms, triangles, &c* as may be mo t convenient^ 
 and measure each figure separately, their sum ^vill 
 be tlie area of the whole polygon. 
 
 As it is presumed the pupil has not made a great 
 progress in arithmetic, the measurement of figures 
 requiring farther calculation is reserved till a future 
 time. It may, however, be not improper to notice 
 that workmen in general reckon three times the dia- 
 meter to be equal to the circumference of a circle, 
 which is not strictly accurate. 
 
 The diameter of the circle may be considered in 
 proportion to its circumference, as 7 is to 22. 
 
 If the diameter be 14, then by the Rule of Three, 
 7 : 22 : : 14 : 44. 
 
 If the circumference be 44, then 22 : 7 :: 44 : It. 
 
 »5 
 
 To Measure a Circle. 
 
 Rule.— Multiply half the circumference by half 
 the diameter — the product will be the area. 
 
49 
 
 SECTION XlV. 
 
 Steredmetrt/^ on the Measuring of Solidi^. 
 
 Agreeably id the plan proposed, the following di-^ 
 rections are given to find the superficies and the solid 
 content of a few of the tnost familiar figures. 
 
 * ^repio^i solidi fJHTpiw^ to measure, 
 A CUBE. 
 
 To Find the Superficies. 
 Ru liE. — Multiply the area of one side by 6. 
 
 To Find the Solid Content* 
 
 Rule. — Multiply the area of one side by the 
 length of one side. 
 
 A PARALLELOPIPEDON, A PRISM, OR A 
 CYLINDER. 
 
 To Find the Superficies. 
 
 Rule. — Add the area df the foitr sides ami th« area 
 of the two ends together. 
 
 To Find the Solid Content. 
 
 Rule. — Multiply the area of one end by the 
 length* 
 
 J 
 
50 
 A CONE OR A PYRAMID. 
 
 To Find the Superficies* 
 
 Rule. — MiiUiplj half the perimeter* of thebase 
 bj tlie length of tiie side. 
 
 If the superficies of the base be also required, find 
 its area and add it to the above. 
 
 *■ TTtpi^ about. fjATpoi'y measure. 
 
 To Find the Solid Content. 
 
 Rule. — Multiply the area of the base hj one third 
 of the perpendicular height. 
 
 A SPHERE. 
 
 To Find the Supeificics of a Sphere. 
 
 Rule. — Multiply the diameter of the sphere by 
 its circumference. 
 
 To Find the Soliditj/. 
 
 Rule. — Multiply the surface by one sixth of lh« 
 diameter. 
 
 The Figures requisite for Forming the Five Solid 
 Bodies,} eommonly called the Platonic Bodies. 
 
 There are only five sorts of regular solids. 
 
 A Tetraedon.f— Fig, 8S. 
 
 Consists of four equilateral triangles, which when 
 folded according to the lines will form a pyramid. 
 
 + TiTpat, four. E/pce, a base. 
 
51 
 
 jifi Ilexaedron, — Fig. 89. 
 
 Consists of six equal squares, ^vhich ^vlien foMed 
 wp ^ill form an liexaedron or cube. 
 
 A 71 Ocicwdro7i, — Fig. 90. 
 Consists of eight equal triangles. 
 
 A Dodecaedron. — Fig. 91. 
 Consists of twelve pentagons. 
 
 An Tcosaedron, — Fig. 92. 
 
 Consists of twenty triangles. 
 
 These figures described on pasteboard, &c. and 
 partially cut along the lines, may be folded till the 
 sides meet, whicli being pasted or glued together, 
 ivill give the form of tlie five Solid Bodies whose 
 sides arc equals and whose surfaces are similar and 
 equal, 
 
 SECTION XV. 
 
 Altimetrj/^ and Longimetri/.f or the Mcnsura- 
 
 tion of Heights, Depths, and Distances. 
 
 To Asccrkdn amj Short Inaccessible Distance^ as iJic 
 Breadth of a River bj/ the Verge of the Hut. — Fig. 93. 
 
 Draw down the verge till its Q{\ge appears to 
 touch the spot whose distance is to b^ measured : 
 ■'^* JltuSy bigb, (jATp^so, measure. 
 
 + Lotigus^ lon^. 
 
 I 2 
 
53 
 
 place one hand under the chin Ip lieep the head 
 steady, then turn towards some level ground, and 
 observe a point in that line in which the verge of tlie 
 hat limits the view — the distance from the observer 
 to that point being measured will give the answer. 
 
 The inaccuracy qf this method is obvious frpm the 
 almost impossibility of keeping the head so steady 
 as neither to enlarge nor to diminish the angle of 
 vision in the act of turning. 
 
 To Measure the Height of an Accessible Object by the 
 Heflection from a Looking-glass on a Bason of 
 Water.— F\g, 94, 
 
 Let a mirrpr qr bason of water, with two lines 
 across to determine the centre, be placed level with 
 the base of the object, and let the observer move back 
 till the top of the object appears at the centre of th^ 
 reflecting surface. 
 
 The height of the object will be in the same pro- 
 portion to the distance of its base from the point of 
 reflection that the height of the eye is to the distance 
 of the observer from the point of reflection. 
 
 Rule. — Multiply the height of the observer's eye 
 by the distance between the mirror and tlie base of 
 the object, and divide by the distance betwepn \h^ 
 mirror an^ thp observer. 
 
53 
 
 Tiius — Tiie lieiglit of liie obseiver's eye • • • • 5i\, 
 The distance of the observer from } g ,^. 
 
 tlic mirror • . . . • 
 
 Distance between the mirror and ^ .cyf. 
 t!ie base of the object ^ •••••• > 
 
 (3 : 5 :: 13 
 
 5 ft in. 
 
 6 )(}rr( 10 10 
 
 J?o Measure the Height of an Accessihle Object^ as a 
 Tree^ Obelisk^ or Tower by the Lejiglh of Us Shaduic^ 
 when either the Sun qt the Moon shine so as to prO'- 
 dure a Shadow. — Fig. 95. 
 
 Place a staff perpciidicularly in the same plane 
 with the tree — measure the height of the staff— (he 
 length of its sliadow^ and of the shadow of the 
 object. 
 
 The shadows of all objects in the same plane 
 being in the same proportion to their respective 
 objects, if tlie staff* be o feet in height and its 
 shadow 2 feet in length , a sliadow 20 feet in length 
 will be produced by a tre^ SO feet Jiigh. 
 
 Rule. — Multiply the shadow of the tree, obelisk^ 
 or tow^er by the length of tjie staff*, and diyide the pror 
 duct by the shadow of the stc^if. 
 
54 
 
 Tliu3— Shadow of ilic Tree SO ft. 
 
 Heiglit of the SlalF S ft. 
 
 Shadow of the Siaft ••...•.. 5 ft. 
 o : 3 :: 80 
 3 
 
 Jnsw.— 48 fL 
 
 To Measure the Height of an Accessible Qhject h/ 
 means of a Slaffindependenllj/ofiis Shadow, — Fig. 96. 
 
 Fix in tl]e ground a staff whose top will be higlicr 
 than tli« eye of the observer — select a station whence 
 the top of the staifand i\ic^ top of the object may be 
 viewed in a direct line. 
 
 Rule. — Multiply the Iieight of the staff ])y tlic 
 distance of the staff from tlie object and divide the 
 product bv the distance of the observer from the 
 staff. 
 
 Thus— The Leiglit of tlie staff oh. 
 
 Distance of t]ie staff from the object 60 ft. 
 Distance of observer 'rom tlic sjaff . . 6 i\. 
 
 To talce the Iieight of an Accessible Object by the 
 Quadrant only. 
 
 Look through ihii, sights to tlie top of (Ijc object, 
 and advance towards it or recede from it, till Xhe 
 plumb line cuts 45^^ then tlic distance of the ()!)server^ 
 from the object will be exactly equal to tlie height of 
 the object. 
 
55 
 
 If the line cut 22|^, the distance of ih^ observer will 
 be twice the height of the object. 
 
 If the line cut Til'', the distance of the observer will 
 be half the height of the object. 
 
 To Measure the Distance of a Thunder Cloud during 
 the Stoj^m, 
 
 Suspend a ball bj a wire or thread S9 inches and 
 ^ tentlis of an inch in length, which will form a pen- 
 dulum vibrating seconds — at ih<i instant tlie flash 
 of lightning is perceived, set the ball in motion and 
 notice (he number of vibrations before the thunder is 
 heard ; and because the velocity of sound is i 142 
 feet in a second, multiply the number of vibrations 
 by 1142, and the product will give the distance of 
 the cloud. 
 
 Thus — If 11 seconds elapsed between seeing the 
 flasli of lightning and liearing the thunder. 
 1142 
 11 
 
 Feet in a mile 5280 ) 12362 ( 2miies667yds. 1 ft. Am, 
 10560 
 
 Feet in a yard 3) 2002 ( 
 667—1 
 
 The length of a half-second pendulum is 9 inches 
 and 3 tentJi«. 
 
 The number of vibrations counted bet ween the flash 
 and the report of a gun will show the distance of a 
 ship, &c. 
 
56 
 
 To A^ctrtain the Vtlocili/ of the Wind injluencrng a 
 Cloud. 
 
 On any levd space notice an extremity of the 
 sliadow of a cloud and observe the fi umber of seconds 
 that elapse during its passage to another point at ii 
 measured distance. 
 
 The number of seconds will be to one hour as the 
 distance of the two places to the fourth number: or 
 
 Dividii the distance by the number of seconds, and 
 it will give the velocity of the Wind per second — mul- 
 tiply by 60 and it will give the velocity of the Wind 
 per minute — multiply by 60 again and it will give the 
 velocity per hour. 
 
 In like manner the velocity of Water may be ascer- 
 tained in a uniform current* 
 
 To Ascer-tain Heights and Depths hy the Velocity * of 
 Falling Bodies, 
 
 Let fall a stone or bullet from the top of the height 
 or depth to be measured, and notice tlie number of 
 second* that elapse during its fall; the velocity of its 
 descent will be continually increasing according to 
 the numbers 1, 3, 5, 7, 9, 1 1, &c. 
 * FelocitaSj swiftness, 
 
 A body will flill 16 feet in the hi second. 
 
 Tiiree times 16, i.e. 48 feet in the 2d second, 
 which added to the former, will be 64 feet in two 
 seconds. 
 
67 
 
 Five times 16, i. e. 80 feet in the 3d second, which 
 tidded to 64 feet, will be 144 (eety the space passed 
 througli in three seconds; 
 
 Seven times 16, 112 feet in the 4th second, which 
 added to 144 feet, will be 256 feet, the space pa»s» 
 ed over in four seconds. 
 
 Nine times 16, i.e. 144 feet in the 5th second, 
 which added to ^56 (eetj will be 400 feetj the space 
 passed over in five seconds. 
 
 Rule. — Multiply the time of descent by its own 
 jQumbef and multiply the product by 16. 
 
 Requihed. — ^How far will a falling body descend 
 in Jive seconds; 
 5 
 5 
 
 25 the square of the time. 
 16 
 
 3)400 feet. . 
 Answ* — 133 yards 1 foot. 
 
 To Ascertain the Altitude of an Object^ as a Tauer^ 
 Steepky Sfc. hy the Angle of Elevation at one 
 station. — Fig. 97, 
 
 At a measured distance, suppose lOO feei^ in a 
 tight line from the object, take with d quadrant the 
 angle of the elevation of the top of the object — for 
 instance 47**^ 
 
58 
 
 Dra"^^ A B 100 — erect a perpendicular at B — lay 
 clown angle A 47°, and continue the line till it inter- 
 sects the perpendicular at C — then B C ^vill be the 
 height of the object* 
 
 Add the height of tlie observer's eye 5 feet 
 Jnsw.—107^. 
 
 To Ascertain the Height of an Accessible or Tnac* 
 cessihle Object hy the Angles of Elevafion at two stations^ 
 a measured distance fro7n each other, — Fig. 98. 
 
 If the object be fixed as a tower ^ an obelisl., &c. 
 one person may make both observ ations ; but if the 
 object be in motion^ as a hite^ a balloon^ cloudy &c. 
 two persons in a right line at a measured distance 
 from each other must take their observations at the 
 same instant. 
 
 liequired, the perpendicular height of a Balloon. 
 
 Angle A 35''^ angle B 55""^ and the distance be- 
 tween A and B 300 feet. 
 
 Draw A D and setoiFA B 300 — lay down angle 
 A 35""^ angle B 55^, continuing the lines till tliey in- 
 tersect each other— from the point of intersection let 
 fall a perpendicular, then C D will be the height re- 
 quired. 
 
 Answ. — 412. 
 
 To Ascertain the Distance of an Accessible or Trnicces" 
 sible Object bi/ the Angle of Depression, — Fig. 99* 
 Required the distance of a ship, IS'' being the 
 
59 
 
 angle of depression when taken from the top of a 
 tower 145 feet in height. 
 
 Draw a line A B to represent the base, erect i he 
 perpendicular A C 145 feet — then draw C D parallel 
 to A B, and draw C B, forming an angle of 18° with 
 the line C D, and C B will be the distance from the 
 top of the tower to the liull of the vessel. 
 
 In like manner the distance from the mast of a 
 ghip to an object on shore or from the mast of one 
 vessel to the hull of another may be ascertained. 
 
 To Ascertain the Height of one Object on the Top of 
 
 another^ as of a Spire above the Tower ^ or of a Castle 
 
 on a Hill or Rock. 
 
 If accessible, on a level with the base of the lower 
 object, at a measured distance take the angle of 
 elevation of the top of the spire or castle, and thien 
 find the height of the lower object and subtract the 
 latter from the former. 
 
 If inaccessible, take the height of each object by 
 ano'les at two stations and subtract as before. 
 
 K 2 
 
60 
 
 SECTION X\ I. 
 
 Miscellanea* Relative to Geography, '\ &;c. 
 
 Incidence:!: & Reflection, § — Whatever angle 
 is formed with a surface by the line described by a 
 body striking that surface, a similar angle Mill fee 
 formed by the substance in its rebound. 
 
 Thus a substance striking in a direction forming 
 fight angles with the surface A B will rebound at 
 right angles — if in ^ny direction more oblique, as 
 K C and F C, it will rebound with a similar angle 
 beyond the point of contact. — Fig. 100. 
 
 Hence will be understood that whatever is the angle 
 oi incidence formed by a ray of light, the angle of re- 
 Jleciion will be the same. 
 
 This may be evidenced by any one viewing ob- 
 jects either directli/ o^ obliquelj/ reflected from a 
 mirror. 
 
 Sound is reflected in a similar manner. 
 
 * Mtffycc^ Miscev*, to mix various things together. 
 + r«, the Earth, yfttpa^ to describe. 
 J Inculo, to fall upon. 
 § ReJleciOy to turn l^ack. 
 
 Kefr ACTION. Ij — Light ig not only reficcted per- 
 pendicularly or obliquely, but it also undergoes ^vh^t 
 
 J Refringo^ to breaks 
 
61 
 
 te termed a refrnciion on passing out of ore medium - 
 into another of a clifTerent degree of density ;+ as 
 from the rarer fluid in the higher regions into tlie 
 .common atmospherict air, and from the air around 
 us into glass or ^vafer, &c. 
 
 Suppose A B io be a body of glass^ water, 
 or oil, a ray of light, as C d, passing through one 
 inediuni and entering another in a perpendicular di- 
 reetion will proceed perpendicularly; but if it enter 
 the second medium obliquely, as r d, the direction of 
 its course will be changed in the denser medium be- 
 coming rnorp inclined toward^ the perpendicular 
 CF— Hg. 101. 
 
 When the ray passes from a denser medium, as 
 glass^ &c. into one more rare, as air^ &g. the refrac- 
 tion causes the ray to proceed in a direction farther 
 from the perpendicular. 
 
 Refraction may be exemplified by placing a shil- 
 ling at the bottom of a bason, and preventing its slip- 
 ping by means of putty or wax, &c. If one person 
 lake a station which admits pf seeing tlie inside of the 
 bason without seeing the shilling— and another pej- 
 json pour water into the bason, the reflected ray 
 /rom the shilling Tyhich before §truck against the side 
 
 * Medium» 
 
 + Aiwu^j densu$y compact, thick. 
 
 I A/^o^, vapour, S^fc/p^'? a sphere. 
 
62 
 
 of the bason becomes refracted and meets the eye of 
 the spectator. — i7>. 102. 
 
 The Sensible* HoRizox.t — The boundary of 
 the view of a spectator on the surface of the earth or 
 water where the eye can have a field of vision iinin- 
 terriipfed till the sky and the land or water appear 
 to unite in a lar^rer cicle, — Tia:* 103. 
 
 * SentWj to perceive. 
 
 f Op/ffe', to bound or terminate. 
 
 The Rational:|: Horizon — is the view whicha 
 spectator might be conceived to have if stationed 
 at the centre of the earth, 
 
 J Ralio^ reason. 
 
 Great as the difTerence appears with respect to the 
 earth, it is inconsiderable as it respects the heavens, 
 because the fixt stars are so remote that the semi- 
 diameter of the eartli about 4000 miles can make no 
 material did'ercncc in their appearance. 
 
 It is evident that' a planet at b would be in the 
 horizon of a spectator at the centre of the earth — at 
 B in the horizon to person on the surface. 
 
 The quantity of the angle B C b is called the hori- 
 zontal Parallax,^ i. e. the arc intercepted between the 
 true and the apparent place of any of the heavenly 
 bodies. 
 
 § IlctpA\hct^i^, diffcreace. 
 
63 
 
 Twilight* is a varying effect produced bj tlie 
 reflection and refraction of light, and gradually iu- 
 creases or decreases in proportion io \\\q Sun's dis- 
 tance below the horizon^ and ceases to be percepti- 
 ble when that distance is more than J8''. 
 
 * Probably from 'twixt aud Hght^ which accords with 
 the French, entre Chien et Loup, 
 
 To take the Altitude of the Sun bj/ a Qundrant, 
 
 If the instrument be either Gunter's or the more 
 homely quadrant mentioned p. 36 — unless it be pro- 
 vided with a dark glass — let the quadrant be held 
 with its face towardi^ the ol)server, so that the Sun'g 
 beams may pass through both sights — then the plumb 
 line will cut the degree expressing the angle of 
 altitude. 
 
 The Sun, when near the horizon, appears higher 
 than its real situation, in consequence of its rays being 
 refracted in the atmosphere. 
 
 The degree noted by the quadrant will not be 
 strictly accurate on account of the refraction which 
 takes place in the atmosphere. The difference be- 
 tween the real and apparent altitude of tlie Sun, or 
 any of the heavenly bodies, is greatest when they are 
 near the horizon ; in consequence of this re/raction 
 the Sun is visible a short time before it rises and, 
 after it sets. 
 
64 
 
 To Draw a Figure of a Globe with StraigJit Lines^ 
 elevaled for the Latitude'^ of LomUri* — Fig. 104. 
 
 With an extent of 60 from the lint of chords, de- 
 scribe a circle — draw two diameters at right angles, 
 HO the horizon, Z N the zenith f and the nadir/ 
 ^Vitli b\\^ taken from the line of chords, place one 
 foot of the compasses on Oand intersect the circle be- 
 tween O & Z — from that point of intersection draw a 
 diameter N P, S P to represent the axisof the^torld, 
 the extremities of which are the North and the South 
 poles. Draw a diameter at riglit angles with the 
 axis to represent the equator, and at 23|° from the 
 pole — draw lines representing (he arcticand antarctic 
 circles, and at ^3\ on each side the equator draw on 
 the North side the tropic of Cancer or summer sol- 
 stice^ and on the South side the tropic of Capricoru 
 or the winter solstice. 
 
 Take with the compasses the extent from O to th» 
 pole, and the same extent will reach from tlue equator 
 to the latitude of the place. 
 
 * Latitude and Longitude are terms used in Geogra* 
 phy. riAct'Ju^, latus^ broad, latitudo. 
 
 Latitude is the distance of any place from the Equator 
 in either a Northern or Southern direction. 
 
 OyKO';, Xoyxp^^ longus^ io^g? longitudo. 
 
 Longitude, the distance of any place in an Eastern or 
 Western direction from any point fixed on. 
 
 + Zenith and Nadir are two Arabic terms adopted in 
 all modern languages. Zenith, the point over head—** 
 Nadir, the point opposite to Zenith* 
 
65 
 
 In like manner in a figure constructecl for any lati- 
 tude the elevation of the pole at that place will be 
 «imilar to the latitude. 
 
 To Find the Latitude of ani/ Place hy the Bleridian'^ 
 AUiLude, — Fig. 105. 
 
 Draw II O to represent i\\e horizon — erect a per- 
 pendicular at the centre C, and with a radius of 60 
 from the line of cliords describe the semi-circle 
 II Z O for the meridian. 
 
 Having found the meridian altitude at the sum- 
 mer solsticet to be 62° — set off62 from H to S. The 
 declination of the sun at that time being 23^"^^ set off 
 S E and the arc fl E will be the altitude of the equa- 
 tor, i.e. the complement (or as abbreviated, co-lati- 
 tude,) being S8|, the remainder of the quadrant 51| 
 will be the latitude of the place. 
 
 In like manner proceed with the meridian altitude 
 at any other time. 
 
 In the summer half year set off the declination 
 above, and in the winter set it oft' below i\m meridian 
 altitude. 
 
 * Meridtes^ noon-day. 
 
 + 5o/, the Sun. Sto^ to stand, because the Sun ap- 
 pears to stand for a time at the solstices. 
 
66 
 
 A Table of the Length in Geographical Miles for 
 each Degree of Longitude according to the Degree of 
 Latitude. 
 
 
 
 60.00 
 
 31 
 
 51.43 
 
 61 
 
 29.09 
 
 1 
 
 .59.99. 
 
 32 
 
 50.88 
 
 62 
 
 28.17 
 
 2 
 
 .^9.96 
 
 S3 
 
 50.32 
 
 &3 
 
 27.24 
 
 S 
 
 59.92 
 
 34 
 
 49.74 
 
 64 
 
 26.30 
 
 4 
 
 59.85 
 
 30 
 
 49.15 
 
 65 
 
 25.36 
 
 5 
 
 59.77 
 
 36 
 
 48.54 
 
 m 
 
 24.41 
 
 6 
 
 .f)9.67 
 
 37 
 
 47.92 
 
 67 
 
 23.44 
 
 7 
 
 59.56 
 
 38 
 
 47.28 
 
 68 
 
 22.48 
 
 8 
 
 ,59.42 
 
 39 
 
 46.63 
 
 69 
 
 21.50 
 
 9 
 
 59.26 
 
 40 
 
 45.96 
 
 70 
 
 20.52 
 
 10 
 
 59.09 
 
 41 
 
 45.28 
 
 71 
 
 19.53 
 
 11 
 
 58,90 
 
 42 
 
 44.59 
 
 72 
 
 18.54 
 
 12 
 
 58.69 
 
 ii3 
 
 4aS8 
 
 73 
 
 17.54 
 
 15 
 
 58.46 
 
 44 
 
 43.16 
 
 74 
 
 16.53 
 
 It 
 
 58.22 
 
 45 
 
 42.43 
 
 75 
 
 15.53 
 
 15 
 
 57.95 
 
 46 
 
 41.68 
 
 76 
 
 14.52 
 
 16 
 
 57.67 
 
 47 
 
 40.92 
 
 77 
 
 13.50 
 
 17 
 
 57.S8 
 
 48 
 
 40.15 
 
 78 
 
 12.47 
 
 18 
 
 57.06 
 
 49 
 
 39.36 
 
 79 
 
 11.45 
 
 19 
 
 56.73 
 
 50 
 
 38.57 
 
 80 
 
 10.42 
 
 20 
 
 5fiS8 
 
 51 
 
 37.76 
 
 81 
 
 9.38 
 
 y] 
 
 56.02 
 
 52 
 
 .S6.94 
 
 82 
 
 8.34 
 
 22 
 
 55.63 
 
 53 
 
 36.11 
 
 83 
 
 7.31 
 
 23 
 
 So.'iS 
 
 54» 
 
 35.27 
 
 84 
 
 6.27 
 
 24 
 
 54.81 
 
 65 
 
 34.41 
 
 ! 85 
 
 5.23 
 
 25 
 
 54,38 
 
 56 
 
 33.55 
 
 ■ m 
 
 4.18 
 
 26 
 
 53.93 
 
 57 
 
 32.68 
 
 1 87 
 
 3.14 
 
 27 
 
 53.46 
 
 58 
 
 31.79 
 
 i 88 
 
 2.0^ 
 
 28 
 
 52.97 
 
 5'> 
 
 30.90 
 
 • 89 
 
 1.04 
 
 2.9 
 
 52.47 
 
 60 
 
 30.00 
 
 1 90 
 
 0.00 
 
 30 
 
 51.96 
 
 
 
 J 
 
 1 
 
67 
 
 To Find the Number of Miles contained in a Degree 
 of Longitude in any given Latitude. 
 
 Draw the line A B, fig. IO65 and divide it into 60 
 geographical or 70 English miles. On A ^vith the 
 extent A B describe an arc, and from B set oif the 
 given iatitnde C (suppose 5^") — let fall a perpen- 
 dicular to D5 and A D will be the measure of a de- 
 gree of longitude in that latitude* 
 
 To Draw a Right Lined Map of a Part of the Earth^s 
 Surfaee, — Fig. 107. 
 
 Suppose it were required to delineate that part of 
 the world which is contained betv^een 54^^ and 59* 
 North latitudcj and between 1° and 7° West longi- 
 tude. 
 
 Draw a line to represent the lower boundary of 
 the map 54° latitude. On the middle of the line 
 erect a perpendicular and divide it, according io the» 
 given number 5 of the degrees of latitude, by lines 
 parallel to the base line. 
 
 Either select a scale equal to the distance of the 
 parallels, or draw one and divide it into six equal 
 parts containing 10 each. 
 
 On each side the perpendicular set off from the 
 scale distances corresponding to the numbers given 
 in the prefixed table — thus, against 54° stands 35. :^7, 
 which would have been 35| had the decimal been 25. 
 
 Therefore on the lower line, i. e. the parallel of 54% 
 
68 
 
 set off tlie extent 35|* three times on each side the 
 
 perpendicular. 
 
 Seek in the table dO^'y it appears 30.90, which ma j 
 
 be considered 31° — that extent on the scale is to be 
 set off on the uppermost line three times on each side 
 
 the perpendicular, i. e. the parallel of 59° — draw the 
 meridian lines to connect the corresponding divi- 
 sions and annex the numbers. 
 
 In this instance the parallels and meridians arc 
 drawn at each degree ; — in maps comprising a larger 
 portion of tlie earth's surface it is usual to draw them 
 at every fifth or every tenth degree. 
 
 To Take an Enlarged or Reduced Cop?/ of any Map^ 
 Draooing or Plan, 
 
 Divide the plan, Szc. and the paper on which the 
 new drawing is to made into an equal number of 
 squares, and delineate in the squares oft he new work 
 4he coFresponding squares of the given drawing. 
 
 To Reduce a Rectilineal Figure. 
 Make a point near the centre of the copy and draw 
 iines from every angle to that point — measure the 
 angles and the lines of the copy — lay down the 
 angles as in the copy and set off the lines according 
 1o a larger or smaller scale, as proposed. 
 
 In preparing to draw a map, the following method, 
 as described by Mr. Bonnycastle, may be found cove- 
 
69 
 
 iiient for erecting perpendiculars, especially near the 
 margin of tlie paper. 
 
 To Raise a Perpendicular from any point B in a given 
 line A. B.—hig. 108. 
 
 1 . From any scale of equal parts take a distance 
 equal to 3 divisions, and set it from B to m. 
 
 2. And from the points B and m, with the distances 
 4 and 5, taken from the same scale, describe arcs cut- 
 ting each other in n. 
 
 3. Through the points n and B draw the line B C, 
 and it will be the perpendicular required. 
 
 The object of the preceding pages has not 
 been to interfere with the subjects of Geogra^ 
 phy or the construction of Maps; the single 
 instance of a Right Lined Projection is intend^ 
 ed to exercise the Pupils in graduating, Sgc. 
 
 The plan usually adopted to impress the 
 figures on the mind, is for the Pupils to dram 
 them, of any size, on cards ; and the mode 
 of examination is to require a verbal explec^ 
 nation of the figures, 
 
 BACON, KIN.n'eBROOK, A.\D CO. PRI.NTERS, NORWICH, 
 
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