Page:Cyclopaedia, Chambers - Volume 1.djvu/945

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5° One Number being given to be divided by another., to find the ghlotient — Extend the Compaffes from the Divi- ior E.gr. 2 5. to i, and the fame Extent will reach from the Dividend E. gr. 750, to the Quotient 305 or, extend the Compaffes from the Divifor to the Dividend, the fame Ex- tent will reach the fame way from 1 to the Quotient.

4 Q 'three Numbers being given, to find a fourth in di- retl 'Proportion — Extend the Compaffes from the firft Num- ber, fuppofe 7, to the fecond, v.g. 14. : that done, the fame Extent applied the fame way from the third, 22, will reach to the fourth Proportional fought, viz. 44.

5 three Numbers being given, to find a fourth in In- Derfe Proportion — Extend the Compaffes from the firft of the given Numbers, fuppofe 60, to the fecond of the fame Denomination, viz. 30 ; ffthe Diftance be applied from the thirdfNumbeT backwards, 5, it will reach to the fourth Num- ber fought, 25.

6 ° three Numbers being given, to find a fourth in duplicate Proportion — If the Denominations of the firft and fecond Terms be Lines, extend the Compaffes from the firft Term to the fecond, of the fame Denomination : This done, that Extent being applied twice the fame way from the third Term, the moveable Point will fall on the fourth Term required. E. gr. the Area of a Circle, whole Diameter is 14, being 154, what will the Content of a Cir- cle be, whofe Diameter is 28 ; applying that Extent the fame way from 1 54 twice, the moveable Point will fall on 616, the fourth Proportional or Area fought.

7 n to find a mean Proportional between two given Num- bers — Biffecf the Diftance between the given Numbers, the Point of Bifection will fall on the mean Proportional fought. Thus the Quotient of the two Extremes divided by one another. Extremes being 8 and 32, the middle Point be- tween them will be found 16.

8 Q to find two mean Proportionals bet-ween two given Lines — Triffect the Space between the two given Extremes; the two Points of Triffection will give the two Means requi- red—Thus if 8 and 27 be the two given Extremes, the two Means will be found 12 and 18.

9° to find the Square Root of any Number under iooooco — The Square Root of a Number is always a mean Propor- between 1, and the Number whofe Root is required; yet with this general Caution, that if the Figures of the dum- ber be even, that is, 2, 4, 6, 8, 10, &c. then you mult look for the Unit at the beginning of the Line, and the Number in the fecond Part or Radius, and the Root in the firft Part ; or rather reckon 10 at the End to be Unity ; and then both Root and Square will fall backwards towards the Middle in the fecond Length or Part of the Line — If they be odd, the Middle 1 will be moft convenient to be counted Unity , and both Root and Square will be found from thence for- wards towards io- — On this Principle the Square Root of 9 will be found to be 3 ; the -Square Root of 64, to be $,&c.

io" to find the Cube Root of any Number tinder tococooooo-The Cube Root is always the firft of two mean Proportionals between I and the Number given, and there- fore to be found by triffefting the Space between them. Thus the Cube Root of 1 7 2 8 will be found 12; the Root of 17280, nearly 20" ; the Root of 172800 almoft >o\

Though the Point on the Lino reprefenting all the fquare Numbers is in one Place, yet by altering the Unit, it pro- duces various Points and Numbers for their refpective Roots.

, The Rule to find this, is to put Dots, or fuppofe them

put over the firft Figure to the Left-hand, the fourth Fi- gure, the fevenrh, and the tenth : If then the latt Dot on the Left-hand fills on the laft Figure, as it does in 1728, the Unit muft be placed at 1 in the Middle of the Line, and the Root, the Square, and the Cube, will all fall for- wards toward the End of the Line.

If it fall on the laft but 1, as in 17280, the Unit muft 'be placed at 1 in the beginning of the Line, and the Cube in fhe fecond Length ; or the Unit may be placed at 10 at the End of the Line ; and then the Root, the Square, and Cube, will all fall backwards, and be found in the fe- cond Part, between the middle and the End of the Line.

p Thus 'will the Cube Root of 8 be found 2 ; that of 27

3 • that of 6"4, 4 ; that of 125, 5 ; that of 2if7, 6, fgc.

'For particular Ufes af GunterVX»z<; in the Meafu- ring of timber, Gauging of VefTcls, &c. See Slieing-

For other Ufcs in Geometry, Trigonometry,^. See Sector, and GuNTEa's-Scale.

are Arches of Circles, but the Hour-Circles all Curves , drawn by means of feveral Altitudes of the Sun for fome particular Latitude every Day in the Tear. See Stereo- graphic and Projection.

The Ufe of this lnttrument is to find the Hour of the Day, the Sun's Azimuths, i£c. and other common Problems of the Globe ; as alfo to take the Altitude of an Object in Degrees.

See its Tlefcription and Ufe more at large under the Ar- ticle Guutcr'$-Qv ADRANT.

Gimme's- Scale, called alfo by Navigators abfolutcly ths Gunter, is a large Plain Scale, with divers Lines thereon; of great VCe in working Queftions in Navigation, ISc. See Scale and Sailing.

On one Side the Scale reprefented Tab. trigonometry, Fig. 35. are the Line of Numbets, marked Numbers; the Line of artificial Sines, marked Sines ; the Line of artifi- cial Tangents, marked Tangents ; the Line of artificial vcr- fed Sines, marked V. S. the artificial Sines of the Rhumbs, marked S. R. the artificial Tangents of the Rhumbs, mark- ed T. R. the Meridian Line in Mercator's Chart, tnark'd Merid. and equal Parts marked E. P.— To which, on the Shorter Scales, of a Foot long, are ufually added Lines of Latitudes, Hours, and Inclinations of Meridians.

On the Backfide of the Scale are the Lines ufually found on a Plain Scale. See Plain-Scale.

The Lines of artificial Sines, Tangents, and Numbers are fo fitted on this Scale, that by means of a Pair of Com- paffes, any Problem, whether in right-lined or fpherical Tri- gonometry, may be folved very expeditioufly, and with to- lerable Exactnefs ; whence the Imlrument becomes ex- tremely ufeful in all Parts of Mathematicks where Trigono- metry-is concerned; as Navigation, Dialling, Aftronomy. See Trigonometry, £f?£.

The fame Lines are alfo occasionally laid down on Ru- lers to Aide by each other ; hence called Sliding-Gunters ; So as to be ufed without Compaffes : but he that under- stands how to ufe them with, may, by what wc have faid of Evcrard's and Cogre/iall's Slidmg-Rules, ufe them with- out. See SLiDiNc-ic.iV/t'.

Ufe of GvnrzR's-Scale.

i Q the Safe of a right-lined 'right-angled triangle being given, 30 Miles, and the oppofite Angle thereto z6 degrees ; to find the Length of the Hypothenufe.

The trigonometrical Canon or Proportion is thus — As ths Sine of the Angle, 20" Deg. is to the Bafe 30 Miles, fo is Ra- dius to the Length of the Hypothenufe — Set one Foot of the Compaffes, therefore, on the 26th Deg. of the Line of Sines ; and extend the other to 30 on the Line of Numbers, and the Compaffes remaining thus opened, fet one Foot on jo Deg. on the End of the Line of Sines, and extend the other on the Line of Numbers : This will give 68 Miles andahalf, for the Length of the Hypothenufe fouoht.

2 the "Bafe of a right-angled triangle being given, 25 Miles, and the Perpendicular 1 5 ; to find the Angle oppofite to the perpendicular.

As the Bafe 25 Miles is to the Perpendicular 15 Miles, fo is Radius to the Tangent of the Angle fought. — Extend the Compaffes, then, on the Line of Numbers, from 15 the Perpendicular given, to 25 the Bafe given; and the fame Extent will reach the contrary way, on the Line of Tan- gents, from 45 Deg. to 31 Deg. the Angle fought.

3 the Safe of a right-angled triangle being given, fup- pofe 20 Miles, and the Angle oppofite to the Perpendicular 50 'Deg. to find the Perpendiculars

As Radius is to the Tangent of the given Angle 50 Deo. fo is the Bafe 20 Miles to the Perpendicular fought — Extend the Compaffes then on the Line of Tangents, from tha Tangent of 45 Deg. to the Tangent of 50 Deg. and the fame Extent will reach on the Line of Numbers the contra- ry way, from the given Bafe 20 Miles, to the required Per- pendicular 23 £- Miles.

Note, The Extent on the Line of Numbers is here ta- ken from 20 to 23- > forwards ; that the Tangent of 50 Deg. may be as far beyond the Tangent of 4; Deg. as its Com- plement 40 Deg. wantl of 45 Deg.

4° the Safe of a right-angled triangle being given, Pup. pofe 3 5 Miles, and the Perpendicular 48 Miles ; to find tit Angle oppofite to the Perpendicular.

As the Bafe 3 5 Miles is to the Perpendicular 48 Miles, fo is Radius to the Tangent of the Angle fought — Extend fhe Compaffes from 3 j, on the Line of Numbers, to 48 ; the fame Extent will reach the contrary way on the Line of Tangents, from the Tangent of 45 Deg. to the Tangent of 56 Deg. 5 Min. or 53 Deg. J5 Min. — To know which of thofe Angles the Angle fought is equal to, confider that the Perpendicular of the Triangle being greater than tf.e Bafe and both the Angles oppofite to the Perpendicular, and the

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