STEREOMETRY,
OR, THE
ART of GAUGING
MADE EASIE
By the HELP of a New
SLIDING RULE

SECT. I.

A Description of the RULE, and of the several Lines upon it, with their Use in some Questions in Arithmetick.

The rule consists of Three Parts, viz. a Rule of 12 Inches long, and two small Scales to slide in it, which may be drawn out, one towards the right Hand, and the other towards the left, till the whole be three Foot long.

    The principal Lines on the Instrument are those commonly known by the Name of Gunter’s-Line, or Line of Numbers, which are here distinguished one from another by certain Letters, set at the end of the Lines towards the right hand.

    Thus the Lines D, are each of them one single Line of Numbers, beginning at the End of the Rule towards the left hand, from thence continued to the other End.

    The Lines A B, and C, are called double Numbers, each being two Lines or Radius’s of Numbers, the first beginning at the left hand, and ending in the middle of the Rule where the second Radius begins, and is from thence continued to the end at the right hand.

    The Line E is call’d Triple Numbers, being three Radius’s of the Numbers; the first beginning at the left hand, and the third ending at the right hand.

    This Triple Line is equal in length to the double Lines, and all to the single Line, for all the five begin and end at the same Point.

    The Line of Numbers is (now) so well known to most Persons, that it may be thought a sufficient Description to have only said that these are such.

    But for as much as this Instrument may be useful to some who (I presume) do not yet know what the Line of Numbers is, I shall therefore endeavour to explain it as followeth.

    The Line of Numbers, is a Line of Geometrical Proportions, divided first into Nine unequal Parts, call’d Primes, which are distinguish’d by Figures, 1, 2, 3, 4, 5, 6, 7, 8, 9; and then each of these Primes are subdivided into ten other parts (according to the same Reason) call’d Tenths, and again each of those Tenths are subdivided, or at least suppos’d to be subdivided into ten other parts, according as the length of the Line will admit; as here the Line D, being about 11 Inches long, each tenth in the first Prime is really subdivided into ten parts, call’d Centesms, but betwixt the figures 2 and 4, each tenth is divided but into five parts, and therefore each of those parts do signifie two Centesms: Again, from the Figure 4 to the end of the Line, every tenth is divided into two parts, each representing five Centesms: Lastly, each of these Centesms is also supposed divided into ten parts, which by some are called Millions; but a Line of this Length will not admit of this last division.

    The Figures (1, 2, 3, 4, 5, 6 &c.) by which the Primes are distinguis’d, are all arbitrary Points, and may any of them represent so many entire Units, Tens, Hundreds or Thousands; or they may also represent so many Tenth, Hundredth, Thousandth, or Ten Thousandth parts of an Unit.

    1. For whole, or intire Units; Let the first Prime or the Fig. 1 at the beginning of the Line D, represent one Unit, then shall all the Figures towards the right hand, (viz. 2, 3, 4, 5, &c. to 10) represent so many Units, and the tenths in each Prime, will be tenth parts, and the Centesms in each of those tenths will be hundredth parts of an Unit.

    Or, let 1 at the beginning of the Line represent 10 Units, then will each Prime forwards represent 10 times so many Units as the Figures express; thus, the Figures 2, 3, 4, 5, &c. will be 20, 30, 40, 50 &c.

    And when one Prime represents 10 Units, every tenth in that Prime will be 1 Unit, and each Centesm in those tenths will be 1 tenth part of Unit.

    Again, Let the first Prime represent 100, then the Figures 2, 3, 4, 5, &c. will represent 200, 300, 400, 500 &c. and therefore 10 at the end will be 1000; and according to this Supposition, 1 tenth in each Prime will be 10 Units, and in those tenths each Centesm will be 1 Unit.

    2. For Decimal Fractions; let 10 at the end of the Line (at D) represent 1, then each Prime towards the left hand will be .1, and in those Primes each tenth will be .01, and these tenths each Centesm will be .001, part of on Unit.

    To make this more plain, draw out the sliding Piece B, till 1 at the beginning of the Line B, stands exactly against 10 at the end of Line A, for then you have a Line of Numbers four times repeated; upon which, let 1 at the beginning of the Line A, represent 1 Unit; then shall 1 in the middle of the said Line be 10; and 10 at the end thereof (or which is all one, at the beginning of the Line B,) represent 100; and by consequence 1 in the middle of the Line B will be 1000, and for the same reason 10 at the end of the said Line (which is also the end of the Rod) will be 10000.

    But keeping the Rule as it now stands let 10 at the end of the fourth Radius (viz. at B.) represent 1, then shall each Prime in the said fourth Radius represent .1, in the third Radius .01, in the second Radius .001, and in the first .0001 part of an Unit.

    So will 2 in the first Radius be .0002, in the second .002, in the third .02, and in the fourth .2 parts.

    The Numbers and Divisions on the Lines being thus explain’d, it will not be difficult to find the Point upon the Line, where any Number given is represented.

    As for Example; Suppose the Numbers were 1895, for the first Figure thereof (viz. 1.) I count 1 at the beginning of the Line D, for the second Figure I count 8 tenths next following (that is 8 of the greater Divisions betwixt 1 and 2.) then from this Point forward I count 9 Centesms for the third Figure, and for the last Figure 5, I count half the next Centesm, so I find the Point ag, will represent 1895, and by the same Rule the Number 1715 will be found at the Point wg; hence observe

    1. That on a Line of this length, only the four first Figures of any Number propos’d can be discover’d; for if the Number given were 189582, it would be represented at the same Point where the former 1895 was found (viz.) at ag.

    2. That all Numbers which after the first Figure have nothing but Cyphers (as 20, 200, 2000, &c.) are all represented at the same Point.

    So the Numbers 20, 200, 2000, are all represented by the Figure 2, at the beginning of the second Prime.

    3. All Numbers consisting of three Figures, and having a Cypher in the middle, are found within the first tenth of that Prime, at which the first Figure of the Number given is found.

    Example. Let the Number given be 308, for the first Figure I count 3 on the Line (which I find at the beginning of the third Prime) now there being a Cypher in the second place, I must not count any of the tenths; but for the last figure 8 I count 8 Centesms, and that is the Point which represents 308.

    4. All Numbers consisting of four places, and having two cyphers in the middle, must be sought betwixt the beginning of the Prime unto which they belong, and the first Centesm of the same Prime; so 4005 being given, the first Figure (viz.) 4, is found at the beginning of the fourth Prime: Now there being Cyphers in the second and third places, I must not count any of the tenths or Centesms; but for the last Figure 5 I estimate 5 Millions, which is about the middle of the first Centesm, and that is the Point where 4005 is represented.

    Note, Decimal Fractions and mixt Numbers are discover’d after the same manner as whole Numbers; for if the first Number 1895, were 18,95 it will be found at the same Point (viz.) at ag; and by what hath been said, it is easie to find what Number is represented at any Point upon the Line, as will appear in the following Questions of Proportion, of which I come now to treat.

    In Arithmetick I shall take notice of two sorts of Proportion, Arithmetical and Geometrical.

    1. Arithmetical Proportion is when divers Numbers, being compared together, retain amongst themselves equal Differences, as these, 2, 4, 6, 8, &c. And this is either continued, as in the Numbers before propounded (which are also called Arithmetical Progression, or a Rank of Numbers Arithmetically proportional;) or discontinued, as in these, 2, 4, 8, 10, or the like.

    2. Geometrical Proportion, is when divers Numbers being compared together, differ amongst themselves according to the same Rate or Reason, as these 2, 4, 8, 16 &c. for here, as 2 is half 4, so 4 is half 8, and 8 is half 16; this is likewise continued, as in those before propounded (which are also called Geometrical Progression, or a Rank of Numbers Geometrically proportional;) or discontinued, as in these, 2, 4, 16, 32; for as 4 is double2, so is 32 double 16, but so is not 16, being compar’d with 4.

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This page was last updated on 29 March 2000.