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ID Title Author Corrected Date of Publication (TCP Date of Publication) STC Words Pages
A60107 A Short treatise of the description of the sector wherein is also shown the great use of that excellent instrument, in the solution of several mathematical problems. Worgan, John, surveyor. 1697 (1697) Wing S3636; ESTC R41684 18,763 47

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this point represents 276. Again suppose I would represent or find the Number 408 on the Line then for the first figure 4 I take that 4 on the Line next the left hand and for the 0 in the second place I must not take any tenths but for the 8 in the third place I count 8 Centesms and it gives that point which represents 408. The greatest Inconvenience of these Lines is this all Numbers above 3 or 4 places cannot precisely be represented unless the Lines are very long As for Decimal Fractions and mixt Numbers they are discovered after the same manner as whole Numbers were For suppose 2. 76 was required to be represented it will be found at the same point and by the same Rule as the whole Number 276 was And therefore by what hath been said it will be easie to find the point where any Number is represented on the Line especially if it be small Next for the Line of Artificial Sines which are only the Logarithms of the Natural and are therefore transfer'd from a Table of Logarithms to the Scale they begin toward the left hand and are numbered towards the right with the Figures 1 2 3 c. to 10 which stands about the middle of the Line and signifie single Degrees after these it runs on with 10 20 30 c. to 90 which stands at the end next the right hand these Divisions are subdivided into 10 equal parts and those again into 10 5 or 2 parts and sometimes not at all according as the length of the Scale will permit So that if betwixt 1 and 2 in the first part it be divided into ten parts one of these parts will represent 6 Minutes and if each of these Divisions be again halv'd each part will represent 3 Min. If the grand Division in the later part be divided into 10 equal parts each division represents a degree and if these are halv'd each division will be 30 Minutes What is said of the Line of Sines the same may be understood of the Line of Tangents whose divisions begin at 1 and run to 10 in the middle of the Line and signifie only single Degrees after 10 it runs on with 20 c. to 45 which stands at the end of the Line from 45 it runs back again to 90 where you begun as the bare inspection of these Lines on any Scale will more fully declare These Lines are of most Excellent use for by them all Questions of Proportion may be solv'd whether Arithmetical or Geometrical But their principal business is the solution of Plain and Spherick Triangles which they do with great speed and exact enough in many Cases and therefore very necessary for proving your Arithmetick and Geometrick Operations The method of working with them is thus In all Proportions you have three terms given to find a fourth Seek out therefore the first term whatever it be viz. Number Sine or Tangent on its like Line that is if it be a Number look for it on the Line of Numbers if it be a Sine seek it on the Line of Sines c. and in that point set your Compasses then extend the other to the 2 d or 3 d term that is extend it to that which is of the same name with the first for either the 2 d or 3 d will be always like the first the same extent laid from the other term the same way will reach to the fourth term required An Example will make it plain Suppose this Proportion was to be wrought as the Sine of 67 d. 30 m. to the Numb 64. So is Radius or Sine of 90 to the 4th Number required Set one foot of your Compasses in the Line of Artificial Sines on the Number 67 d. 30 m. and extend the other foot to 90 on that Line the same extent will reach from 64 on the Line of Numbers to 70 on the said Line if apply'd the same way which 70 is the 4th Number required After the same manner may all other Questions be wrought CHAP. V. Of the Sector and the Description of the Lines thereon placed HAving thus shown you how these Lines are originally divided I shall now pass to show how they are plac'd on a Sector or joynt Rule and then shall give a few Examples of their Excellent use as they are thus disposed A Sector as 't is Geometrically defined is a Figure bounded by two strait Lines and part of the Circumference of a Circle as Fig. 5. But by a Sector here spoken of you are to understand an Instrument that opens upon a Center like a common Carpenter's Rule The two pieces that moves upon the Center we call Legs upon which are plac'd most of the Lines that are upon the Common Scales but are here used after a different manner than they are upon those Scales The principal Lines that are now generally put upon this Instrument to be used Sectorwise are equal Parts Chords Sines Tangents Secants and Polygons The Line of equal parts called also the Line of Lines is a Line divided into 100 equal parts and if the length of the Leg of the Sector will permit is again subdivided into Halves and Quarters they are placed on each Leg of the Sector on the same side and are Numbered by 1 2 3 c. to 10. which is very near the end of each Leg these Lines are as in the Printed Plate of the Sector hereunto annexed noted with the Letter L. And here note that this 1 may be taken for 10 or for 100 1000 10000 c. as occasion requires and then 2 will signifie 20 200 2000 20000 c. and so of the rest The Line of Chords is a Line divided after the usual way of the Line of Chords from a Circle whose Radius is nearly the length of one of the Legs this Line is placed upon each Leg of the Sector beginning at the Center and running towards the end thereof It is numbered with 10 20 c. to 60 and to this Line on each Leg is set the Letter C. Note this Line on some Sectors run to 90 degrees The Line of Sines is a Line of common natural Sines such as we have before defined only 't is divided from a Circle of the same Radius that the Line of Chords was these are also plac'd upon each Leg of the Sector and numbred with the Figures 10 20 30 c. to 90 at the end of which on each Leg in the Print annexed is set the letter S. The Line of Tangents is a Line of common Tangents divided from a Circle of the forementioned Radius and is plac'd upon each Leg and runs to 45. It has the Numbers 10 20 c. to 45 placed upon it with the Letter T for Tangent Beside this is another small Line of Tangents divided from a Radius of about 2 Inches and is placed upon each Leg of the Sector it begins at 45 which stands at the length of the Radius
set the Fig. 10 20 30 c. as they were in the Arch. Fifthly If the Extents D 10 D 20 D 30 c. in the Tangent-Line be transfer'd to the Line D B produc'd they will divide it into a Line of Secants which with the Numbers 10 20 30 c. may from the Scheme be put upon any Scale and is continued on the line of Sines for after the Sine of 90 the Tangents begins Hence you see that the Secant of Nothing is the Radius or Semi-diameter of the Circle Sixthly If from A to each Division of the Arch B C you draw Lines they shall divide the Radius B D into a Line of half Tangents to which set the Numbers 10 20 30 c. to 90. Sometimes this Line is made to run as far as 150 or 160 Deg. which is of the same length with the whole Tangent of 75 or 80 Deg. and when 't is so divided the Line B D must be produc'd as it was for the Secant For the half Tangent of 10 Deg. is the whole Tangent of 5 Deg. so also the half Tangent of 160 Degrees is the whole Tangent of 80 Degrees Seventhly Draw the Line A F and divide the Arch A F into 8 equal parts these transfer'd from the Arch to the strait Line A F as were the Chords divides it into a Line of Rhumbs or Points Its use is to lay of any Point of the Compass which it doth much more readier than a Line of Chords though that will serve when you have not this Lastly You may divide the Semi-diameter F D into 100 equal parts and it will serve for a Line of Equal parts to take Leagues or Miles from not but that it might be longer or shorter for there is no necessity of its being just of such a length This Line is of great use because from it all the other Lines might be divided by supposing the Radius of the Circle to contain any number of equal parts as 1000 10000 c. For then all the Chords Sines Tangents c. will consist of a certain number of those parts CHAP. III. Of the Plain Scale ASsume a Line whose length let be what you please suppose ab On this Line make a Parallelogram of any breadth at pleasure as abcd divide the opposite sides ab and de each into 10 equal parts as also ad bc seting thereto the Figures 1 2 3 c. as you see in Fig. 4. This done draw from the Points 1 2 3 c. in the Line ad the several Lines parallel to ab then from the Points b 1 2 3 c. in the Line ab to the Points 1 2 3 c. in the Line dc draw the Diagonal Lines b. 1 1. 2 2. 3 3. 4 c. By which the Line ab or those equal to it are actually divided into 100 equal parts The reason of which is very clear from the 4th of the 6th of Euclid for by that proposition 't wil hold as bc be 1 c he and therefore if be is 1 10 of bc which by construction it is then shall he be 1 10 of 1 c that is 1 10 of 1 10 or 1 100 of dc or ab If the Sides ab dc as also their Parallels 1 2 3 c. be produced and the distances ab dc repeated thereon 1 2 3 or more Lines as the lengths of the Rule on which you put them will permit you have finished the said Diagonal Scale the design of which is to supply the use of a Scale of equal parts but with greater exactness than any of those kind of Scales can pretend to for from it may be taken any Number under 1000 exactly CHAP. IV. Of the construction of the Lines of Artificial Numbers Sines and Tangents BEsides the Lines before described there are another sort which is generally put on Scales and Rules called Artificial Numbers Sines and Tangents the rise and construction of which are from the Logarithmes For they are only the Logarithms of the Natural Numbers Sines and Tangents Transfer'd to a Scale The Line of Numbers commonly known by Name of Gunter's Line is a Line of Geometrical Proportion unequally divided into 9 parts beginning at 1 towards the left hand and running with 2 3 4 c. to 10 which is about the middle of the Line where another Radius begins and is continued to 100 towards the right hand it hath the Figures 10 20 30 c. set thereto and is divided just as was the first part of it The Line being thus divided is called a Line of Numbers of two Radiuses and the way of Numbring on it thus for herein lyes the great difficulty in the use of it It is as before was noted divided into 18 unequal parts the first 9 ending about the middle of the Scale and the other 9 at the end next the right hand then each of these Primes or first Grand Divisions are subdivided into 10 other parts according to the same reason called Tenths and again each of those Tenths are subdivided into 10 other parts if the length of the Rule or Scale will permit The Figures 1 2 3 c. by which the Primes are distinguished are all arbitrary points and may each of them represent so many entire Unites Tenns Hundreds or Thousands or they may also represent so many Tenth Hundredth Thousandth or Ten Thousandth parts of an Unite If the first 1 on the Line be taken for 1 then 10 in the middle of the Line is 10 as in course it falls but if the first 1 be taken for 10 or 100 then the figure 10 in the middle of the Rule must be taken for 100 or 1000. Now when the first 1 or Prime represents 10 Unites each Tenth in that Prime will be 1 and each Centesme in those Tenths if there be any will be one tenth part of an Unite Again let the first Prime represent 100 then the Figures 2 3 4 c. will denote 200 300 400 c. and so 10 at the end will be 10000 and according to this supposition 1 tenth in each Prime will be 10 Unites and in those tenths each Centesme will be 1. For Decimal Fractions let 10 at the end of the Line next the right hand represent 1 then each Prime towards the left hand will be 1 and in those Primes each tenth will be .01 and in these tenths each Centesm will be .001 part of an Unite The Divisions and way of Numbring on this Line being thus explain'd it will not I presume be difficult to find the point upon the Line where any Number given is represented For Example Suppose the Number 276 were proposed for the first figure 2 I account 2 next the beginning of the Line that is 2 next the left hand for the second figure 7 I tell 7 tenths next following that is 7 of the 10 great Divisions betwixt 2 and 3 then from this point I count 6 Centesms So that from 2 to