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to nine of the Divifions at the End of the Scale; this is 49; take your Compaffes from the Scale, go down four of the Lines which are drawn parallel to the Sides of the Scale, and there place one Foot, and the other will fall a little short of the Line it before touched at the End of the Scale; extend the Compaffes 'till they touch that Line, and this Extent is 59.4: Now place one Foot in A, with the other cross the Perpendicular BC in C, and draw the Hypothenufe A C, which completes the Triangle ABC.

The given Parts are marked with, and the Part required with o, whether they are Sides or Angles.

Now I obferve, that the Hypothenufe A C is given, and oppofite to the right Angle at B, which, in a right-angled Triangle, is always known from the Triangle; and that the Base A B is given, and is oppofite to the Angle at C, which is one of the Parts required to be found; which is done by the following Proportion according to the general Rule:

As the Hypothenuse A C, 59.4,

1.773786

Is to the Radius, or Sine of the Angle at B, 10.
So is the Bafe AB, 47-5,

1.676694

11.676694

1.773786

To the Sine of the Angle at C, 53°. 6',

9.902908

The fecond and third Numbers being added together, and the firft fubtracted from their Sum, leaves 9.902908 for the Sine of the Angle at C; against which, among the Sines in the Tables of Sines and Tangents, you will find 53°.6', for the Angle at C; which Angle being meafured by the Scale of Chords, by Prob. 9, if it is found to meafure right, it proves that the Conftruction and Calculation are both true.

The Angle at B is
Subtract the Angle at C

Remains the Angle at A

90°.00'
53. 06

36. 54,

36. 54, which may be measured (in the fame Manner as was the Angle at C.

Now I obferve, that the Bafe A B is given, and oppofite to the Angle C, which was found by the former Proportion; and that the Angle A is alfo found, being the Complement of the Angle C to 90°, and is oppofite to the Perpendicular,

the

the other Part required to be found; which is done by the following Proportion :

As the Sine of the Angle at C, 53°. 6'′,

Is to the Base A B, 47.5,

So is the Sine of the Angle at A, 36°. 54,

9.902919

[blocks in formation]

1.552230

To the Perpendicular B C, 35.66,

The fecond and third Numbers being added together, and the firft fubtracted from their Sum, there remains 1.552230 for the Logarithm of the Perpendicular BC. Now, looking for this Logarithm in the Table of Logarithms, neglecting the Index 1, the nearest Logarithm to it is .551400; against which, in the Column on the left Hand, I find 356; and, among the Figures at the Tops of the Columns, over that where my Logarithm is, I find 6, which, being placed to the right Hand of the 356, makes 3566 for the Number anfwering to the Logarithm in the Operation. But now, the Index of the Logarithm being only 1, therefore of the 3566 there must be only two Places Integers; whence 1 conclude 35.66 is the Length of the Perpendicular BC: And, to measure this from the diagonal Scale, place one Foot of the Compaffes at A, and extend the other to B; which Extent will reach on the diagonal Scale from where the fixth Parallel croffes the Divifion marked 3, to the fifth of the fmall Divifions at the End, and fomewhat more, which fhews it to be more than 35.6, as by the Calculation.

C

Prob. 19. In the right-angled Triangle ABC, right-angled at B, given the Perpendicular BC 35.66, and the Angle at A 36°54', to find the Hypothenufe AC, and the Bafe A B. Conftruction. Draw the Bafe-Line A B at Pleasure, on the End of which, by Prob. 3, erect the Perpendicular BC to any convenient Length: Set one Foot of the Compafles on the diagonal Scale at 3, and extend the other to 5 at the End of the Scale; take your Compaffes from the Scale, and go down fix of the

I

A

B

parallel

parallel Lines, and extend them fo as to touch the Point, where this Line, and the Line marked 5 at the End of the Scale, interfect; this is 35.6, which fet off on the Perpendicular from B to C: Then, as the Angle at A is 36°54', the Angle at C, being the Complement of it to 90°, is 53°.6'; take 60° from the Line of Chords, and fet one Foot of the Compaffes in C, and draw a Portion of an Arch; then take 53°.6' from the Line of Chords, and, fetting one Foot of the Compaffes where the Arch cuts the. Perpendicular, with the other interfect the Arch: A Line drawn from C, thro' the Interfection, 'till it cuts the Bafe in A, completes the Triangle.

Here we may obferve, that BC is given, and oppofite to the given Angle at A; and that the Angle B is always known in a right-angled Triangle; and that this is oppofite to the Hypothenufe A C, one of the Parts required; which may be found by the following Proportion :

As the Sine of the Angle at A, 36°.54',

Is to the Perpendicular BC, 35.66,

So is the Radius, or Sine of the Angle at B,

To the Hypothenuse A C, 59.4,

9.778455

1.552241

10.

11.552241 9.778455

1.773786

To measure the Hypothenufe, put one Foot of the Compaffes on the diagonal Scale at 5, and extend the other to nine of the Divifions at the End of the Scale; take the Compaffes from the Scale, go down four of the parallel Lines, and extend them to the Point where this Line croffes the ninth Diagonal at the End of the Scale, and this will be the Measure of the Hypothenufe, if the Work is true.

Here we are to obferve, that the Angle A is known, and oppofite to the Perpendicular, which was given; and that the Angle C is known, and oppofite to the Bafe A B, the other Part required; which may be found by the following Proportion:

As

As the Sine of the Angle at A, 36°. 54',

9.778455

Is to the Perpendicular B C, 35.66,
So is the Sine of the Angle at C, 53°.6',

1.552241

9.902919

11.455160

9-778455

To the Bafe A B2 47.5

1.676705

To measure the Bafe A B, fet one Foot of the Compaffes on the diagonal Scale at 4, and extend the other to the seventh Diagonal at the End; take the Compaffes from the Scale, go down to the fifth Parallel, and place one Foot there, and extend the other to the Point where the fifth Parallel croffes the feventh Diagonal at the End of the Scale; this is the Measure of the Base. And here the Reader may obferve the following

General Rule for taking off any Number from the diagonal Scale.

If it confifts of two Places, put one Foot of the Compaffes on the Scale, to that Number which ftands first to the left Hand in the given Number, and extend the other Foot to the other Number at the End of the Scale; this is the Extent required.

Example. To take 78 from the Scale, fet one Foot of the Compaffes to 7, and extend the other to the Diagonal marked 8 at the End of the Scale; this Extent is 78, the given Number.

If the Number confifts of three Places, fet one Foot of the Compaffes on the Scale, to the first Figure to the left Hand, and extend the other Foot to the fecond Figure of the given Number at the End of the Scale, as before; take the Compaffes from the Scale, and go down the parallel Lines, 'till you come to the Line that has the fame Figure against it at the End with the laft Figure to the right Hand in your given Number; here place one Foot of the Compaffes, and extend the other to where this Parallel croffes the Diagonal at the End of the Scale which is marked with the fecond Figure of your given Number; this is the Extent required.

Example.

Example. To take 946 from the Scale, fet one Foot of the Compaffes to 9, and extend the other to the Diagonal marked 4 on the Edge at the End of the Line; take the Compaffes from the Scale, and go down to the Parallel marked 6 at the End; fix one Foot of the Compaffes on this Parallel, and extend the other to where the Diagonal marked 4 croffes the Parallel; this is the true Extent for the given Number.

Here it is to be obferved, that both the Diagonals and Parallels, for Want of Room, have only the Numbers 2, 4, 6, 8; fo that 1, 3, 5, 7, and 9, are not marked.

Prob. 20. In the right-angled Triangle ABC, right-angled at B, given the Hypothenufe AC 59.4, and the Perpendicular BC 35.66, to find the Bafe AB, and the Angles at A and C.

Conftruction. Draw the Bafe-Line A B at Pleasure, and on the Point B. erect the Perpendicular B C, of a proper Length; then take the given Length 35.6 from the Scale, by the general Rule, Prob. 19, and fet it from B to C: Take, by the fame Rule,

C

B

the Hypothenufe 59.4 from the Scale, and, with one Foot of the Compaffes in C, interfect the Bafe-Line A B in A; then draw the Line A C, and the Triangle is completed.

Here the Hypothenufe is given, and oppofite the right Angle; and the Perpendicular BC is alfo given, which is oppofite the Angle A, one of the Parts required; which may be found by the following Proportion:

As the Hypothenufe A C, 59.4,

1.773786

Is to the Radius, or the Sine of the Angle at B, 10.
So is the Perpendicular B C, 35.66,

1.552241

[blocks in formation]

To the Sine of the Angle at A, 36°. 54,

9.778455

The Angle at A may be measured by Prob. 9. Now, the Angle at A being found, and oppofite to the given Perpendicular B C, the Bafe A B, oppofite to the Angle

Ff

C 53'

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