is 13 deg 30 min. PROBL. 8. The declination given to find the beginning and end of twilight or day-break Lay the thread to the declination counted the contrary way as in the last Problem and take from your Scale of altitudes 18 deg for twilight and 13 deg for day-break or clear light with this run one point of the Compasses along the line of houres on that side next the end until the other will just touch the thread and then the former point gives the respective times required Ex. gr At 7 deg North declination day breaks 8 minutes before 4 but twilight is 3 houres 12 minutes in the morning or 8 hours 52 minutes afternoon PROBL. 9. The declination and altitude of the Sun or any Star given to find their Azimuth in Northern declination Lay the thread to the altitude numbred on the limb of the moveable piece from 60 0 toward the end and when occasion requires continue your numbring forward upon the loose piece and take the declination from your line of altitude with this distance run one point of your Compasses along the line of Azimuths on that side the thread next the head until the other just touch the thread then the former point gives the Azimuth from South Ex. gr at 10 deg declination North and 30 deg altitude the Azimuth from South is 64 deg 40 min. PROBL. 10. The Suns altitude given to find his Azimuth in the aequator Lay the thread to the altitude in the limb counted from 60 0 on the loose piece toward the end and on the line of Azimuths it cuts the Azimuth from South Ex. gr at 25 deg altitude the Azimuth is 53 deg At 30 deg altitude the Azimuth is 41 deg 30 min. fere PROBL. 11. The declination and altitude of the Sun or any Star given to find the Azimuth in Southern declination Lay the thread to the altitude numbred on the limb from 60 0 on the moveable piece toward the end and take the declination from the Scale of altitudes then carry one point of your Compasses on the line of Azimuths on that side the thread next the end until the other just touch the thread which done the former point gives the Azimuth from South Ex. gr at 15 deg altitude and 6 deg South declination the Azimuth is 58 deg 30 min. PROBL. 12. The declination given to find the Suns altitude at East or West in North declination and by consequent his depression in South declination Take the declination given from the Scale of altitudes and setting one point of your Compasses in 90 on the line of Azimuths lay the thread to the other point on that side 90 next the head on the limb it cuts the altitude counting from 60 0 on the moveable piece Ex. gr at 10 deg declination the altitude is 12 deg 40 min. PROBL. 13. The declination and Azimuth given to find the altitude of the Sun or any Star Take the declination from the Scale of altitudes set one point of your Compasses in the Azimuth given then in North declinanation turn the other point toward the head in South toward the end and thereto laying the thread on the limb you have the altitude numbring from 60 0 on the moveable piece toward the end Ex. gr At 7 deg North declination and 48 deg Azimuth from South the altitude is 35 deg but at 7 deg declination South and 50 deg Azimuth the altitude is onely 18 deg 30 min. PROBL. 14. The altitude declination and right ascension of any Star with the right ascension of the Sun given to find the hour of the night Take the Stars altitude from the Scale of altitudes and laying the thread to his declination in the limb find his hour from the last Meridian he was upon as you did for the Sun by Probl. 5. If the Star be past the South this is an afternoon hour if not come to the South a morning hour which keep Then setting one point of your Compasses in the Suns right ascension numbred upon the line twice 12 or 24 next the outward ledge on the fixed piece extend the other point to the right ascension of the Star numbred upon the same line observing which way you turned the point of your Compasses viz. toward the head or end With this distance set one point of your Compasses in the Stars hour before found counted on the same line and turning the other point the same way as you did for the right ascensions it gives the true hour of the night Ex. gr The 22 of March I find the altitude of the Lions heart 45 deg his declination 13 d. 40 min. then by Probl. 5. I find his hour from the last Meridian 10 houres 5 min. The right ascension of the Sun is 46 m. of time or 11 d. 30 m. of the Circle the right ascension of the Lions heart is 9 hour 51 m. fere of time or 147 deg 43 m. of a circle then by a line of twice 12 you may find the true hour of the night 7 hour 13 min. PROBL. 15. The right ascension and declination of any Star with the right ascension of the Sun and time of night given to find the altitude of that Star with his Azimuth from South and by consequent to find the Star although before you knew it not This is no more than unravelling the last Problem 1 Therefore upon the line of twice 12 or 24 set one point of your Compasses in the right ascension of the Star extending the other to the right ascension of the Sun upon the same line that distance laid the same way upon the same line from the hour of the night gives the Stars hour from the last Meridian he was upon This found by Probl. 5. find his altitude as you did for the Sun Lastly having now his declination and altitude by Probl. 8. or 10. according to his declination you will soon get his Azimuth from South This needs not an example By help of this Problem the Instrument might be so contrived as to be one of the best Tutors for knowing of the Stars PROBL. 16. The altitude and Azimuth of any Star given to find his declination Lay the thread to the altitude counted on the limb from 60 0 on the moveable piece toward the end setting one point of your Compasses in the Azimuth take the nearest distance to the thread this applyed to the Scale of altitudes gives the declination If the Azimuth given be on that side the thread toward the end the declination is South when on that side toward the head its North. PROBL. 17. The altitude and declination of any Star with the right ascension of the Sun and hour of night given to find the Stars right ascension By Probl. 5. or 14. find the Stars hour from the Meridian Then on the line twice 12 or 24 set one point of your Compasses in the Stars hour thus found and extend the other to the hour of the night Upon the same line
Styles height above the Substile As the radius is to the cosine of the latitude so is the cosine of declination to the sine of the styles height To finde the Substyles distance from the Meridian As the radius is to the sine of declination so is the co-tangent of the latitude to the tangent of the substyle from the Meridian To finde the angle of twelve and six As the co-tangent of the latitude is to the radius so is the sine of declination to the co-tangent of six from twelve To finde the inclination of Meridians As the sine of the latitude is to the radius so is the tangent of declination to the tangent of inclination of Meridians 6. All North decliners with centers have the angular point of the style downward and all South have it upward 7. All North decliners without centers have the narrowest end of the style downward all South have it upward 8. In all decliners without centers take so much of the style as you think convenient but make points at its beginning and end upon the substyle of your paper draught and transmit those points to the substyle of your plain for direction in placing your style thereon 9. In all North decliners the Meridian or inclination of Meridians is the hour line of twelve at mid-night in South decliners at noon or mid-day This may tell you the true names of the hour lines 10. In transmitting these Dials from your paper draught to your plain lay the horizontal of your paper draught upon the horizontal line of the plain and prick off the hours and substyle Sect. 5. The affections of direct reclining Plains inclining Plains For South Recliners and North Incliners 1. The difference 'twixt your co-latitude and the reclination inclination is the elevation or height of the style 2. When the reclination inclination exceeds your co-latitude the contrary pole is elevated so much as the excess Ex. gr a North recliner or South incliner 50. d. in lat 52. 30. min. the excess of the reclination inclination to your co-latitude is 12. d. 30. min. and so much the North is elevated on the recliner and the South pole on the incliner 3. When the reclination inclination is equal to the co-latitude it s a polar plain For South incliners and North recliners 1. The Sum of your co-latitude and the reclination inclination is the styles elevation 2. When the reclination inclination is equal to your latitude it s an equinoctial plain and the Dial is no more than a circle divided into 24. equal parts having a wyer of any convenient length placed in the center perpendicular to the plain for the style 3. When the reclination inclination is greater than your latitude take the summe of the reclination inclination of your co-latitude from 180. and the residue or remain is the styles height But in this case the style must be set upon the plain as if the contrary pole was elevated viz. These North recliners must have the center of the style upward and the South incliners have it downward Note In all South re-in-cliners North re-in-cliners for their delineation the styles height is to be called the co-latitude and then you may draw them as erect direct plains for South or North as the former rules shall give them in that latitude which is the complement of the styles height For direct East and West recliners incliners 1. In all East and West recliners incliners you may refer them to a new latitude and new declination and then describe them as erect declining plains 2. Their new latitude is the complement of that latitude where the plain stands and their new declination is the complement of their reclination inclination But to know which way you are to account this new declination remember all East and West recliners are North-East and North-West decliners All East and West incliners are South-East and South-West decliners 3. Their new latitude and declination known you may by Sect 4. par 5. finde the substyle from the Meridian height of the style angle of twelve and six and inclination of Meridians using in those proportions the new latitude and new declination instead of the old 4. In all East and West recliners incliners with centers the Meridian lies in the horizontal line of the plain in such as have not centers its paralel to the horizontal line Sect. 6. The affections of declining reclining Plains inclining Plains The readiest way for these is to refer them to a new latitude and new declination by the subsequent proportions 1. To finde the new Latitude As the radius is to the cosine of the plains declination so is the co-tangent of the reclination inclination to the tangent of a fourth ark In South recliners North incliners get the difference 'twixt this fourth ark and the latitude of your place and the complement of that difference is the new latitude sought If the fourth ark be less than your old latitude the contrary pole is elevated if equal to your old latitude it s a polar plain In South incliners North recliners the difference 'twixt the fourth ark and the complement of your old latitude is the new latitude If the fourth ark be equal to your old co-latitude they are equinoctial plains 2. To finde the new declination As the radius is to the cosine of the reclination inclination so is the sine of the old declination to the sine of the new 3. To finde the angle of the Meridian with the Horizontal Line of the Plain As the radius is to the tangent of the old declination so is the sine of reclination inclination to the co-tangent of the angle of the meridian with the horizontal line of the plain This gives the angle for its scituation Observe in North incliners less than a polar the Meridian lyes above That end of the Horizontall Line contrary to the Coast of Declination   below  South recliners more than a polar the Meridian lyes below That end of the Horizontall next the Coast of Declination   above  North recliners less than an equinoctial the Meridian lyes above That end of the Horizontalnext the Coast of Declination   below In North recliners this Meridian is 12. at midnight  equal to an equinoctial the Meridian descends below the Horizontal at that end contrary to the coast of Declination and the substyle lies in the hour line of six   South incliners more than an equinoctial the Meridian lyes below That end of the horizontal contrary to the declination   above In South incliners this Meridian is only useful for drawing the Dial and placing the substyle for the hour lines must be drawn through the center to the lower side After you have by the former proportions and rules found the new latitude new declination the angle and scituation of the meridian your first business in delineating of the Dial will be both for such as have centers and
line of seconds continued unto 60 and marked at the end Se. Next to the outward edge on the fixed and movable piece which is best discerned when those pieces are opened to the full length is a line of Meridians divided to 85 whose use is for Navigation in describing Maps or Charts c. In the vacant spaces you may have a line of chords sines and tangents to any Radius the space will bear and what other any one thinks best of as a line of latitudes and hours c. On the proportional side of the loose piece are lines for measuring all manner of solids as Timber Stone c. likewise for gaging of Vessels either in Wine or Ale measure On the outward ledge of the movable and fixed piece both which in use must be stretched out to the full length is a line of artificial numbers sines tangents and versed sines The first marked N the second S the third T the fourth VS On the inward ledge of the movable piece is a line of 12 inches divided into halfs quarters half-quarters Next to that is a prick'd line whose use is for computing of weight and carriages Lastly a line of foot measure or a foot divided into ten parts and each of those subdivided into ten or twenty more On the inward ledge of the loose piece you may have a line of circumference diameter square equal and square inscribed There will still be requisite sights a thread and plummet And if any go to the price of a sliding Index to find the shadow from the plains perpendicular in order to taking a plains declination and have a staff and a ballsocket the Instrument is compleated with its furniture Proceed we now to the uses Onely note by the way that Mr. Brown hath for conveniency of carrying a pair of Compasses Pen Ink and Pencil contrived the fixed piece and movable both to be hollow and then the pieces that cover those hollows do one supply the place of the loose piece for taking altitudes the other being a sliding rule for measuring solids and gaging Vessels without Compasses CHAP. II. Some uses of the quadrantal side of the Instrument PROBL. 1. To find the altitude of the Sun or any Star HAng the thread and plummet upon the pin at the beginning of the line of sines on the fixed piece and having two sights in two holes parallel to that line raise the end of the fixed piece toward the Sun until the rayes pass through the sights but when the Sun is in a cloud or you take the altitude of any Star look along the outward ledge of the fixed piece until it be even with the middle of the Sun or Star then on the limb the thread cuts the degree of altitude if you reckon from 0 60 on the loose piece toward the head of the movable piece PROBL. 2. The day of the Moneth given to find the Suns place declination ascensional difference or time of rising and setting with his right ascension The thread laid to the day of the Moneth gives the Suns place in the line of signs reckoning according to the order of the Moneths viz. forward from March the 10th to June then backward to December and forward again to March 10. In the limb you have the Suns declination reckoning from 60 0 on the movable piece towards the head for North toward the end for South declination Again on the line of right ascensions the thread shews the Suns right ascension in degrees or hours according to the making of your line counting from Aries toward the head and so back again according to the course of the signs unto 24 hours or 360 degrees Lastly on the line of hours you have the time of Sun rising and setting which turned into degrees for the time from six gives the ascensional difference Ex. gr in lat 52. deg 30 min. for which latitude I shall make all the examples The 22 day of March I lay the thread to the day in the Moneths and find it cut in the Signs 12 deg 20 min. for the Suns place on the limb 4 deg 43 min. for the Suns declination North. In the line of right ascensions it gives 46 min. of time or 11 deg 30 min. of the circle Lastly on the line of hours it shews 28 min. before six for the Suns rising or which is all 7 deg for his ascensional difference PROBL. 3. The declination of the Sun or any Star given to find their amplitude Take the declination from the scale of altitudes with this distance setting one point of your Compasses at 90 on the line of Azimuth apply the other point to the same line it gives the amplitude counting from 90 Ex. gr at 10 deg declination the amplitude is 16 deg 30 min. at 20 deg declination the amplitude is 34 deg PROBL. 4. The right ascension of the Sun with his ascensional difference given to find the oblique ascension In Northern declination the difference betwixt the right ascension and ascensional difference is the oblique ascension In Southern declination take the summ of them for the oblique ascension Ex. gr at 11 deg 30 sec. right ascension and 6 deg 30 sec. ascensional difference In Northern declination the oblique ascension will be 5 deg in Southern 18 deg PROBL. 5. The Suns altitude and declination or the day of the Moneth given to find the hour Take the Suns altitude from the Scale of altitudes and laying the thread to the declination in the limb or which is all one to the day in the Moneths move one point of the compasses along the line of hours on that side the thread next the end until the other point just touch the thread then the former point shews the hour but whether it be before or after noon is left to your judgment to determine Ex. gr The 22 day of March or 4 deg 43 min. North declination and 20 deg altitude the hour is either 47 minutes past 7 in the morning or 13 minutes past 4 afternoon PROBL. 6. The declination of the Sun or day of the Moneth and hour given to find the altitude Lay the thread to the day or declination and take the least distance from the hour to the thread this applyed to the line of altitudes gives the altitude required Ex. gr The 5 day of April or 10 deg declination North at 7 in the morning or 5 afternoon the altitude will be 17 deg 10 sec. and better PROBL. 7. The declination and hour of the night given to find the Suns depression under the horizon Lay the thread to the declination on the limb but counted the contrary way viz. from 60 0 on the movable piece toward the head for Southern and toward the end for Northern declination This done take the nearest distance from the hour to the thread and applying it to the line of altitudes you have the degrees of the Suns depression Ex. gr at 5 deg Northern declination 8 hours afternoon the depression
with this distance set one point of your Compasses in the right ascension of the Sun and turning the other point the same way as you did for the hour it gives the Stars right ascension PROBL. 18. The Meridian altitude given to find the time of Sunrise and Sunset Take the Meridian altitude from your particular Scale and setting one point of your Compasses upon the point 12 on the line of hours that is the pin at the end lay the thread to the other point and on the line of hours the thread gives the time required PROBL. 19. To find any latitude your particular Scale is made for Take the distance from 90 on the line of Azimuth unto the pin at the end of that line or the point 12 this applyed to the particular Scale gives the complement of that latitude the Instrument was made for PROBL. 20. To find the angles of the substile stile inclination of Meridians and six and twelve for exact declining plains in that latitude your Scale of altitudes is made for Sect. 1. To find the distance of the substile from 12 or the plains perpendicular Lay the thread to the complement of declination counted on the line of Azimuths and on the limb it gives the substile counting from 60 0 on the moveable piece Sect. 2. To find the angle of the Stile 's height On the line of Azimuths take the distance from the Plains declination to 90. This applyed to the Scale of altitudes gives the angle of the stile Sect. 3. The angle of the Substile given to find the inclination of Meridians Take the angle of the substile from the Scale of altitudes and applying it from 90 on the Azimuth line toward the end the figures shew the complement of inclination of Meridians Sect. 4. To find the angle betwixt 6 and 12. Take the declination from the Scale of altitudes and setting one point of your Compasses in 90 on the line of Azimuths lay the thread to the other point and on the limb it gives the complement of the angle sought numbring from 60 0 on the moveable piece toward the end This last rule is not exact nor is it here worth the labour to rectifie it by another sine added sith you have an exact proportion for the Problem in the Treatise of Dialling Chap. 2 Sect. 5. Paragr 4. CHAP. III. Some uses of the Line of natural signs on the Quadrantal side of the fixed piece PROBL. 1. How to adde one sign to another on the Line of Natural Sines TO adde one sine to another is to augment the line of one sine by the line of the other sine to be added to it Ex. gr To adde the sine 15 to the sine 20 I take the distance from the beginning of the line of sines unto 15 and setting one point of the Compasses in 20 upon the same line turn the other toward 90 which I finde touch in 37. So that in this case for we regard not the Arithmetical but proportional aggregate 15 added to 20 upon the line of natural sines is the sine 37 upon that line and from the beginning of the line to 37 is the distance I am to take for the summe of 20 and 15 sines PROBL. 2. How to substract one sine from another upon the line of natural sines The substracting of one sine from another is no more than taking the distance from the lesser to the greater on the line of sines and that distance applyed to the line from the beginning gives the residue or remainer Ex. gr To substract 20 from 37 I take the distance from 20 to 37 that applyed to the line from the beginning gives 15 for the sine remaining PROBL. 3. To work proportions in sines alone Here are four Cases that include all proportions in sines alone CASE 1. When the first term is Radius or the Sine 90. Lay the thread to the second term counted on the degrees upon the movaeble piece from the head toward the end then numbring the third on the line of sines take the nearest distance from thence to the thread and that applyed to the Scale from the beginning gives the fourth term Ex. gr As the Radius 90 is to the sine 20 so is the sine 30 to the sine 10. CASE 2. When the Radius is the third term Take the sine of the second term in your Compasses and enter it in the first term upon the line of sines and laying the thread to the nearest distance on the limb the thread gives the fourth term Ex. gr As the sine 30 is to the sine 20 so is the Radius to the sine 43. 30. min. CASE 3. When the Radius is the second term Provided the third term be not greater than the first transpose the terms The method of transposition in this case is as the first term is to the third so is the second to the fourth and then the work will be the same as in the second case Ex. gr As the sine 30 is to the radius or sine 90 so is the sine 20 to what sine which transposed is As the sine 30 is to the sine 20 so is the radius to a fourth sine which will be found 43 30 min. as before CASE 4. When the Radius is none of the three terms given In this case when both the middle terms are less than the first enter the sine of the second term in the first and laying the thread to the nearest distance take the nearest extent from the third to the thread this distance applyed to the scale from the beginning gives the fourth Ex. gr As the sine 20 to the sine 10 so is the sine 30 to the sine 15. When only the second term is greater than the first transpose the terms and work as before But when both the middle tearms be greater than the first this proportion will not be performed by this line without a paralel entrance or double radius which inconveniency shall be remedied in its proper place when we shew how to work proportions by the lines of natural sines on the proportional or sector side These four cases comprizing the method of working all proportions by natural sines alone I shall propose some examples for the exercise of young practitioners and therewith conclude this Chapter PROBL. 4. To finde the Suns amplitude in any Latitude As the cosine of the Latitude is to the sine of the Suns declination so is the radius to the sine of amplitude PROBL. 5. To finde the hour in any Latitude in Northern Declination Proport 1. As the radius to the sine of the Suns declination so is the sine of the latitude to the sine of the Suns altitude at six By Probl. 2. substract this altitude at six from the present altitude and take the difference Then Proport 2. As the cosine of the latitude is to that difference so is the radius to a fourth sine Again Proport 3. As the
the fourth term sought The work is by Probl. 2. If 60 men do a work in 5 dayes how long will 30 men be about it As 30 is to 60 so is 5 to 10. The number of dayes for 30 men in the work Again when the fourth term is to be less than the single denomination say As the greater double denomination is to the lesser double denomination so is the single denomination to the fourth term sought If 30 men do a work in 5 dayes how long shall 60 be doing of it As 60 is to 30 so is 5 to 2½ The time for 60 men in the work PROBL. 4. The length of any perpendicular with the length of the shadow thereof given to finde the Suns Altitude At the length of the shadow upon the lines of lines is to the Tangent 45 so is the length of the perpendicular numbred upon the lines of lines to the tangent of the Suns altitude PROBL. 5. To finde the Altitude of any Tree Steeple c. at one station At any distance from the object provided the ground be level with your Instrument look to the top of the object along the outward ledge of the fixed piece and take the angle of its altitude This done measure by feet or yards the distance from your standing to the bottom of the object Then say As the cosine of the altitude is to the measured distance numbred upon the lines of lines so is the sine of the altitude to a fourth number of feet or yards according to the measure you meeted the distance to this fourth adde the height of your eye from the ground and that sum gives the number of feet or yards in the altitude CHAP. VI. How to work proportions in Numbers Sines or Tangents by the Artificial Lines thereof on the outward ledge THe general rule for all of these is to extend the Compasses from the first term to the second and observing whether that extent was upward or downward with the same distance set one point in the third term and turning the other point the same way as at first it gives the fourth But in Tangents when any of the terms exceeds 45 there may be excursions which in their due place I shall remove PROBL. 1. Numeration by the Line of Numbers The whole line is actually divided into 100 proportional parts and accordingly distinguished by figures 1 2 3 4 5 6 7 8 9 and then 10 20 30 40 50 60 70 80 90 100. So that for any number under 100 the Figures readily direct you Ex. gr To finde 79 on the line of numbers count 9 of the small divisions beyond 70 and there is the point for that number Now as the whole line is actually divided into 100 parts so is every one of those parts subdivided so far as conveniency will permit actually into ten parts more by which means you have the whole line actually divided into 1000 parts For reckoning the Figures impressed 1 2 3 4 5 6 7 8 9 to be 10 20 30 40 50 60 70 80 90 and the other figures which are stamped 10 20 30 40 50 60 70 80 90 100 to be 100 200 300 400 500 600 700 800 900 1000. You may enter any number under 1000 upon the line according to the former directions And any numbers whose product surmount not 1000 may be wrought upon this line but where the product exceeds 1000 this line will do nothing accurately Wherefore I shall willingly omit many Problems mentioned by some Writers to be wrought by this line as squaring and cubing of numbers c. Sith they have only nicety and nothing of exactness in them PROBL. 2. To multiply two numbers given by the Line of Numbers The proportion is this As 1 on the line is to the multiplicator so is the multiplicand to the product Ex. gr As 1 is to 4 so is 7 to what Extend the Compasses from the first term viz. I unto the second term viz. 4. with that distance setting one point in 7 the third term turn the other point of the Compasses toward the same end of the rule as at first and you have the fourth viz. 28. There is only one difficulty remaining in this Problem and that is to determin the number of places or figures in the product which may be resolved by this general rule The product alwayes contains as many figures as are in the multiplicand and multiplicator both unless the two first figures of the product be greater than the two first figures in the multiplicator and then the product must have one figure less than are in the multiplicator and multiplicand both Ex. gr 47 multiplied by 25 is 2175 consisting of four figures but 16 multiplied by 16 is 240 consisting of no more than three places for the reason before mentioned I here for distinction sake call the multiplicator the lesser of the two numbers although it may be either of them at pleasure PROBL. 3. To work Division by the Line of Numbers As the divisor is to 1 so is the dividend to the quotient Suppose 800 to be divided by 20 the quotient is 40. For As 20 is to 1 so is 800 to 40. To know how many figures you shall have in the quotient take this rule Note the difference of the numbers of places or figures in the dividend and divisor Then in case the quantity of the two first figures to the left hand in your divisor be less than the quantity of the two first figures to the left hand in your dividend the quotient shall have one figure more than the number of difference But where the quantity of the two first figures of the divisor is greater than the quantity of the two first figures of the dividend the quotient will have only that number of figures noted by the difference Ex. gr 245 divided by 15. will have two figures in the quotient but 16 divided 〈…〉 â—â—ve only one figure in the quotient PROBL. 4. To finde a mean proportional 'twixt two Numbers given by the line of Numbers Divide the space betwixt them upon the line of numbers into two equal parts and the middle point is the mean proportional Ex. gr betwixt 4 and 16 the mean proportional is 8. If you were to finde two mean proportionals divide the space 'twixt the given numbers into three parts If four mean proportionals divide it into five parts and the several points 'twixt the two given numbers will show the respective mean proportionals PROBL. 5. To work proportions in Sines alone by the Artificial Line of Sines Extend the Compasses from the first term to the second with that distance set one point in the third term and the other point gives the fourth Only observe that if the second term be less than the first the fourth must be less 〈…〉 or if the second term exceed 〈…〉 fourth will be greater than the third This may direct you in all proportions of sines and tangents singly or conjunctly to
in length makes a Foot As the square in feet and inches on the right is to one foot on the left so is the point Sq. on the right to the number of feet and inches on the left for a foot square of Timber At 18 inches square 5 inches ¼ and almost half a quarter in length makes a foot When your Timber if it be proper to call such pieces by that name is under 3 inches square account the figured divisions on the right line from the letters Sq. to the end for inches and each sub-division twelve parts of an inch So that every three of them makes a quarter of an inch Then the proportion is as the inches and quarters square on the right is to 100 on the left so is the point 12 'twixt D and T D on the right to the number of feet in length on the left to make a foot of Timber As 2 inches ½ square you must have 23 foot 6 inches and somewhat better for the length of a foot of Timber PROBL. 3. The square and length of a plece of Timber given to finde the content As the point Sq. on the right is to the length in feet and inches on the left so is the square in feet and inches on the right to the content in feet on the left At 30 foot in length and 15 inches square you have 46 foot ½ of Timber At 20 foot in length and 11 inches square you have 16 foot and almost ¼ of Timber When you have a great piece of Timber exceeding 100 foot which you may easily see by the excursion upon the rule then take the true square and half the length sinde the content thereof by the former proportion and doubling that content you have the whole content PROBL. 4. The Circumference or girth of a round piece of Timber being given together with the length to finde the content As the point R. on the right is to the length in feet and inches on the left so is the circumference in feet and inches on the right to the content in feet on the left At 20 foot in length and 7 foot in girth you have 60 foot of Timber for the content This is after the common allowance for the waste in squaring and although some are pleased to quarrel with the allowancer as wronging the seller and giving the quantity less than in truth it is yet I presume when they buy it themselves they scarcely judge those Chips worth the hewing and have as low thoughts of the over-plus as others have of that their admonition If it be a Cylinder that you would take exact content of then say As the point T R on the right is to the length on the left so is the girt on the right to the exact content on the left At 15 foot in length and 7 foot in girt you have 59 foot of solid measure The Diameter of any Cylinder given you may by the same proportion finde the content placing the point D instead of R in the proportion for the usual allowance and the point T D for the exact compute PROBL. 5. To measure tapered Timber Take the square or girt at both ends and note the sum and difference of them Then for round Timber as the point R. on the right is to the length on the left so is half the sum of the girt at both ends on the right to a number of feet on the left Keep this number and say again As the point R. on the right is to the third part of the former length on the left so is half the difference of the girts on the right to a number of feet on the left which number added to the former gives the true content The same way you may use for square Timber only setting the feet and inches square instead of the girt and the point Sq. instead of the point R. At 30 foot in length 7 foot at one end and 5 at the other in girt half the sum of the girts is 6 foot or 72 inches the first number of feet found 67 half the difference of the girts is 1 foot or 12 inches the third part of the length 10 foot then the second number found will be 7 foot one quarter and half a quarter The sum of both or true content 74 foot one quarter and half a quarter For standing Timber take the girt about a yard from the bottom and at 5 foot from the bottom by Chap. 7 set down these two diameters without the bark and likewise the difference 'twixt them Again by Chap. 6. Probl. 4. finde the altitude of the tree so far as it bears Timber or as we commonly phrase it to the collar this done you may very near proportion the girt at the collar and content of the tree before it falls In case any make choice of the hollow contrivance mentioned Chap. 1. they need no compasses in the mensuration of any solid provided the lines for solid measure and gauging vessels be doubly impressed only in a reverted order one pair of lines proceeding from the head toward the end and the other pair from the end toward the head upon the sliding cover and its adjacent ledges This done the method of performing any of the Problems mentioned in this Chapter is easie For whereas you are before directed to extend the Compasses from the first term to the second and with that distance setting one point in the third term the other point gave the fourth or term sought So here observing the lines as before slide the cover until the first term stand directly against the second then looking for the third on its proper line it stands exactly against the fourth term or term sought on the other line Only note that when the second term is greater than the first it s performed by that pair of lines proceeding from the head toward the end But when the first term is greater than the second it is resolved by that pair of lines which is numbred from the end toward the head CHAP. X. The Gauge Vessels either for Wine or Ale Measure PROBL. 1 The Diameter at Head and Diameter at Boung given in Inches and tenth parts of an Inch to finde the mean Diameter in like measure TAke the difference in inches and tenth parts of an inch between the two diameters Then say by the line of numbers As 1 is to 7 so is the difference to a fourth number of inches and tenth parts of an inch This added to the Diameter at head gives the mean Diameter Ex. gr At 27 inches the boung and 19 inches two tenths at the head the difference is 7 inches 8 tenths The fourth number found by the proportion will be 5 inches 4 tenths and one half which added to the diameter at the head gives 24 inches 6 tenths and one half tenth of an inch for the mean diameter PROBL. 2. The length of the Vessel and the mean Diameter given in Inches and tenth parts of
which end of the rule to turn the point of your Compasses for finding the fourth term Ex. gr As the sine 60 is to the sine 40 so is the sine 20 to the sine 14. 40. Again As the sine 10 is to the sine 20 so is the sine 30 to the sine 80. PROBL. 6. To work proportions in Tangents alone by the Artificial Line of Tangents For this purpose the artificial line of tangents must be imagined twice the length of the rules and therefore for the greater conveniency it is doubly numbred viz. First from 1 to 45 which is the radius or equal to the sine 90 In which account every division hath as to its length on the rule a proportional decrease Secondly it s numbred back again from 45 to 89 in which account every division hath as to its length on the line a proportional encrease So that the tangent 60 you must imagine the whole length of the Rule and so much more as the distance from 45 unto 30 or 60 is This well observed all proportions in tangents are wrought after the same manner of extending the Compasses from the first term to the second and that distance set in the third gives the fourth as was for sines and numbers But for the remedying of excursions sith the line is no more than half the length we must imagine it I shall lay down these Cases CASE 1. When the fourth term is a tangent exceeding 45 or the Radius Ex. gr As the tangent 10 is to the tangent 30 so is the tangent 20 to what Extending the Compasses from 10 on the line of tangents to 30 with that distance I set one point in 20 and finde the other point reach beyond 45 which tells me the fourth term exceeds 45 or the radius wherefore with the former extent I set one point in 45 and turning the other toward the beginning of the line I mark where it toucheth and from thence taking the distance to the third term I have the excess of the fourth term above 45 in my compass wherefore with this last distance setting one point in 45 I turn the other upon the line and it reacheth to 50 the tangent sought CASE 2. When the first term is a tangent exceeding 45 or the Radius Ex. gr As the tangent 50 is to the tangent 20 so is the tangent 30 to what Because the second term is less than the first I know the fourth must be less than the third All the difficulty is to get the true extent from the tangent 50 to 20. To do this take the distance from 45 to 50 and setting one point in 20 the second term turn the other toward the beginning of the line marking where it toucheth extend the Compasses from the point where it toucheth to 45 and you will have the same distance in your Compasses as from 50 to 20 if the line had been continued at length unto 89 tangents with this distance set one point in 30 the third term and turn the other toward the beginning because you know the fourth must be less and it gives 10 the tangent sought CASE 3. When the third term is a Tangent exceeding 45 or the Radius As the tangent 40 is to the tangent 12 40 min. so is the tangent 65 to what Extend the Compasses from 40 to 12 d. 40 min. with distance setting one point in 65. turn the other toward 45 and you will finde it reach beyond it which assures you the fourth term will be less than 45. Therefore lay the extent from 45 toward the beginning and mark where it toucheth take the distance from that point to 65 and laying that distance from 45 toward the beginning it gives 30 the tangent sought These Cases are sufficient to remove all difficulties For when the second term exceeds the Radius you may transpose them saying as the first term is to the third so is the second to the fourth and then it s wrought by the third Case I suppose it needless to adde any thing about working proportions by sines and tangents conjunctly sith enough hath been already said of both of them apart in these two last Problems and the work is the same when they are intermixed Only some proportions I shall adjoyn and leave to the practice of the young beginner with the directions in the former Cases PROBL. 7. To finde the Suns ascensional difference in any Latitude As the co-tangent of the latitude is to the tangent of the Suns declination so is the radius to the sine of the ascensional difference PROBL. 8. To finde at what hour the Sun will be East or West in any Latitude As the tangent of the latitude is to the tangent of the Suns declination so is the radius to the cosine of the hour from noon PROBL. 9. The Latitude Declination of the Sun and his Azimuth from South given to finde the Suns Altitude at that Azimuth As the radius to the cosine of the Azimuth from south so is the co-tangent of the latitude to the tangent of the Suns altitude in the equator at the Azimuth given Again As the sine of the latitude is to the sine of the Suns declination so is the cosine of the Suns altitude in the equator at the same Azimuth from East or West to a fourth ark When the Azimuth is under 90 and the latitude and declination is under the same pole adde this fourth ark to the altitude in the equator In Azimuths exceeding 90 when the latitude and declination is under the same pole take the equator altitude out of the fourth ark Lastly when the latitude and declination respect different poles take the fourth ark out of the equator altitude and you have the altitude sought PROBL. 10. The Azimuth Altitude and Declination of the Sun given to finde the hour As the cosine of declination is to the sine of the Suns Azimuth so is the cosine of the altitude to the fine of the hour from the Meridian Proportions may be varied eight several wayes in this manner following 1. As the first term is to the second so is the third to the fourth 2. As the second term is to the first so is the fourth to the third 3. As the third term is to the first so is the fourth to the second 4. As the fourth term is to the second so is the third to the first 5. As the second term is to the fourth so is the first to the third 6. As the first term is to the third so is the second to the fourth 7. As the third term is to the fourth so is the first to the second 8. As the fourth term is to the third so is the second to the first By thesse any one may vary the former proportions and make the Problems three times the number here inserted Ex. gr To finde the ascensional difference in Problem 10 of this Chapter which