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Once Upon A Time ?

Once Upon a Time, Hi Everyone,I am Sure this will bring back some memories,and if you never seen it before ,then take a Gander,because there was a time when you had to know and do this,but using coordinates will work also,I will get around to that.And it has been along time since i have even sit down and done this.I will leave out Balancing  Latitudes and Departures, and just go straight to it,But that does affect the answers as well and decimal places used, when comparing Area to a calculator and computer programs.But there was once upon a time that did a lot of math,not only with calculators,but also in there heads.I still have a Slide Rule, a log log, made by Pickett, those were the Days,                No calculator.just a slide rule and some books.full of tables

I was talking with Charlie Aycock, one day on equations used to convert Lat. and Long. coordinates to Lambert coordinates and Mercator, asked by Omar Lozada, I told Charlie that i just showed how to do this on a HP 35s calculator with programs by Dr Bill Hazelton  he also shows all these equations,Not that we had not showed some ,because we had,but i commented  on using your calculator as i always do.and how important using one is.Then i here from Doug Tarrant ,this comment ,"And there was a time when relying on a calculator for the answer without doing some mental maths was considered bad practice".then i comment back hi Doug,you must be a Dino like me,And yes it is true ,you did here comments ,like you can't add 1+1,ever time you pick up a calculator,and i recalled to him about doing all this math,on pencil and paper and every thing we had to do converting HMS to Decimal or vice versa i then mentioned DMD and DPD .So take a Gander, and see what it was like,and like i said i left out some of it,and i am sure some of you old timers will smile.And Kudos to you Doug Tarrant,because have come along way from Slide Rules to computers.and i got your 6 any day. So here is Once Upon a Time.

Double Departure's or DmD's....Double Meridian Distance and Double Parallel Distance.
Problem #1..DMD & PDP...........+N-S...............+E-W.....Clock Wise Rotation
Bearing's........Distance's.........LAT's.............DEP's.............DD or DMD's..Formula or AREA.
or AZ's


14.20-00.........44.50...................43.11............11.02..................11.02.......................... 475.07

81.40-00.........54.75.....................7.94............54.17..................76.21........................+ 605.11

169.49-00.......69.65.................- 68.55............12.31................142.69.....................+ -9781.40

270.00-00.......39.10.....................0.00............-39.10................115.90............................+ 0.00

294.30-00.......42.20....................17.50..........-38.40..................38.40........................+ 672.00

..........................................................total algebraic = - 8029.22 / 2 =4014.61 m sq

.................DMD =d DEP Sums , 11.02..........................AREA =DMD x LAT

..................+11.02+54.17 =76.21...................................11.02 x 43.12 = 475.07

.........76.21 +54.17+12.31 =142.69...................................76.21 x 7.94 = 605.11

........142.69 +12.31-39.10 = 115.90..............................142.69 x -68.55 = -9781.40

........115.90 -39.10 -38.40 =38.40....................................115.90 x 0.00 =0.00

......................................................................................38.40 x 17.50 =672.00

.............................. total algebraic = - 8029.22/ 2 =4014.61 m sq.√ no adj.balance to trav

...............................adj. on calculator program =4015.0594 m sq.run @ 4dec.pl.for CK. √

DPD.....Double Parallel Distance......+E-W....DEP...........+N-S....LAT..................PDP............AREA
.................Trapezoids Areas.....................11.02....................43.11.................43.11........475.07
...............................................................54.17.....................7.94...................94.16........5100.65
...............................................................12.31.................. -68.55..................33.55.........413.00
............................................................ .-39.10...................0.0000................-35.00........1368.50
............................................................. -38.40..................17.50..................-17.50........672.00
...........................................................total algebraic = 8029.22 /2=4014.61 m sq √

.......................DPD =D LAT.....................................AREA =DPD x DEP......
...................................=.43.11................................11.02 x 43.11 = 475.07

.......................+43.11+7.94 =94.16..........................54.17 x 94.16 = 5100.65

............+94.16+7.94+ -68.55=33.55.........................12.31 x 33.55 = 413.00

.........33.55+ -68.55+0.0000 = -35.00..................... .- 39.10 x -35.00 = 1368.50

........-35.00+0.0000+17.50 = -17.50......................... -38.40 x -17.50 = 672

.........................................................total algebraic =8029.22 /2 =4014.61 m sq √


Now for one Acre of Area @0° to 90° in sqft,,Counter Clock Wise Rotation
problem#2


Bearings............Distance................lat's.............Dep's..............DD or DMD's...Formula or AREA

N-90.00-00E.........208.7103............0.0000..........208.7103............208.7103......................0.0000

N-0.-00-00-E........208.7103.........208.7103...........0.0000...............417.4206..................+ +87,119.9787

N-90.-00-00-W......208.7103............0.0000...........-208.7103............208.7103....................+0.0000

S-0.-00-00W.........208.7103.........208.7103...........0.0000................0.0000......................+0.0000.
...................................................AREA =DMD x LAT........total algebraic=+87,119.9787 sqft
..................DMD =DEP Sums.....=208.7103......................0.0000 x 208.7103 =0.000
...................+208.7103+0.0000=417.4206.........................208.7103 x 417.4206 = 87,119.9787
.....417.4206+0.0000+ -208.7103=208.7103
.....208.7103+ -208.7103 +0.0000=0.0000..............................0.0000 x 208.7103 =0.0000
.........................................................................................- 208.7103 x0.0000 =0.0000 ..........................................................................................total......................=87,119.9787sqft
.............................................................................................87,119.9787/2=43,559.9893 sqft
.............................................................................................43,559.9893/43560 =1.0000 AC.
.............................................................................................208.7103 x² =43599.9893 sqft

Now one Acre of Area@45°to 45° in sqft ,Clock Wise Rotation
Problem#3


Bearings.................Distance.......................lat's...........Dep's..........DD or DMD.......Formula or AREA

N-45.-00-00E........208.7103..................147.5805.........147.5805.........147.5805.............21,780.0040

S-45.-00-00-E.......208.7103................- 147.5805.......  147.5805..........442.7415......+ - 65,340.0119

S-45.-00-00-W......208.7103................. -147.5805...... -147.5805.........442.7415.......+ -65,340.0119

N-45.-00-00-W......208.7103.................  147.5805........- 147.5805.......147.5805......... +21,780.0040
...................................................................................................total...algebraic= -87,120.0158 sqft


.......................DMD =DEP Sums......=147.5805................................AREA=DMD x LAT
...................... +147.5805 +147.5805 =442.7415...................147.5805 x147.5805 =21,780.0040
............442.7415 +147.5805 + -147.5805=442.7415............ -147.5805 x 442.7415 = -65,340.0119
..........442.7415+ -147.5805 + -147.5805 =147.5805........... - 147.5805 x 442.7415 = -65,340.0119
.........................................................................................147.5805 x 147.5805 = 21,780.0040
...........................................................................................total algebraic = - 87,120.0158 sqft
..........................................................................................87,120.0158 /2=43560.0079 sqft
..........................................................................................43560.0079 /43560.00 = 1.0000 AC.
..........................................................................................208.710326 x² =43560.000000

Notice the Clock wise and Counter Clock Wise Rotations are Noted in solutions.

In modern and engineering,area calculations are seldom every done this way,by hand,but let it be said that many a time a wish that a hour or two had been spent to check computer area calculations by hand or on a calculator and now days both,when there area is reported wrong on a Plat or any document .Nothing is perfect. And think about this very time you think you had a hard day and all the math you did not have to do,So remember Once Upon a Time.

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  • HI Everyone, Just wanted to comment on ,when calculating deflections , using a division factional formulas. take care of you digital grouping and your number Mantissa of how your calculator rounds a number off in Decimals of a Degree.It will appear you have some wrong ,when in fact you do not. A example we will use, what is being used in this curve problem. Running the calculator in fix 4, you will see 4 decimals places. the calculator  is rounding your numbers off. Example lets take the Degree of Curve 25.40-20 HMS,change it to DD's, 25.6722, your real 25.6722 is 25.67222222222 / that by 200 = your answer is 0.1284. Your real answer, is 0.128361111111 this is what you don't see, you see 0.1284. so in doing the multiplications if you are not using the Mantissa Number you answer will  be different.

    Example 14.8770 x 0.1284= 1.9102, that is 1.54-37 HMS.  Now using the Mantissa numbers,

    same distance, 14.8770 x 0.128361111111 = 1.9096, that is 1.54-35 HMS, that is little difference.

    Because you are using a small distance. Lets jump to the end of the Curve, and see what happens .

    length =177.2585 x 0.1284 = 22.7600 DD,now  Mantissa Numbers 177.2585 x 0.1283611111111 =22.7531.

    DD  that is 22.45-11 vs, 22.45-12 1/2 Delta angle, this is because of a fraction of a second.

    Now the answer of 22.7600  that is 22.45-36 HMS ,this is a big different from 14.8700 feet to 177.2585 ft.

    This is 24 seconds off the 1/2 Delta angle, this is the different this makes when using 1/200  per foot.

    Using the Mantissa  or the input number at fix 4,you can see the difference.

    Deflections for 100 ft of arc is D/2, so 25.40-20/2=25.6722 =12.8361, remember this number,

    14.8700 x 12.8361 /100=1.9096, now 177.2585 x 12.8361 /100= 22.7531.  I done this above using, 

     length / 100 x 1/2 D = Deflection angle. this is length  x D /100 = deflection angle . So 1/200 will work ,but you must use Mantissa numbers. on factional numbers. when using that small of a faction.

    Another equation i like  i showed above also is deflections = DFA= ( 90 x L ) ÷ ( Pi x R ) I have used this a lot. Just thought i go over all of this, so no one  would be puzzle  about their answers.

    degree.it
  • Hi Everyone,  Still talking about Deflection Angles. In my Text. file on deflection angles i talk about different fractions as in Ft,/100 x (1/2 D) = Deflections, for a 100 ft of arc.  100/100 =1 x D is the Degree at 100 feet of arc. So 100 ft. = ( 1/2 D ) this formula can be broken down like i showed 50 ft of arc = ( 1/4 D )  or (D/4),25 ft of arc = ( 1/8 D ) or ( D/8) and 1 ft of arc  = ( 0 1/200 D ) or ( D / 200 ) next lets look at my curve.

    The PC is at Sta.=6+85.1230 there is 14.8770 arc ft. to 7+00, Using this equation is of 1 ft of arc = D /200.

    Degree in this curve is 25.40-20 HMS that is 25.6722 DD / by 200= 0.1284 ,now take that and multiply it

    by the arc distance of you stations, from the PC to the Station. As like this.

    7+00 =14.8770 x 0.1284 = 1.9096 or 1.54-35 HMS

    7+50 =64.8770 x 0.1284 = 8.3277 or 8.19-40 HMS

    8+00 =114.8770 x 0.1284 =14.7457 or 14.44-45 HMS

    8+50 =164.8770 x 0.1284 = 21.1638 or 21.09-50 HMS

    8+62.3815 =177.2585 x 0.1284 = 22.7531 or 22.45-11 vs.22.45-12 this is decimal placement 0.5+/- sec faction. This can happen in any deflection angle calculation or in any Decimal conversion  to HMS.

    so there is no error. Just like the Decimal  angle of 53.13 is = to 53.1301 same as 36.87 = 36.8699.

    So there you have it one more way to calculate  Deflections. With out a program. Just your calculator. Now  also here is one more equation for  I central angle,  I already gave you Two ,but here is one more,

      DA or I = D x L / 100  i already gave you this one, very similar, I = L / 100 x D

     and that is how it was done Once Upon a Time, not long ago. And it still can be done today.

  • Hi Everyone, I final got time to finish, explaining deflection angle Layout. So here you are ,just you. your calculator and your Total station and your crew. This method may require at lease 3 or 4 persons with a Transit. Although, It can be done with Two , but double work walking back and forth . So its better with 3 on a big curve, Total station No problems,2. Below i will attach a text. file. This is too much to Paste, It will be on how to layout a simple curve, and how it was calculated for this staking. So this is how it was done Once Upon a Time. So when you see and think about this, think about those 4 Presidents carved into that Mountain. Three of them were Surveyors, and the surveying they done in their life. Because they did it their way, Once Upon a Time .

    • For those of you who have never done this, It's not as hard as it looks ,but for those who have. I bet this will bring back some survey memories on those curves, of days gone by.

  • thank you Mr.Billy I want to study it again

  • Hi Everyone, To do any calculation using angles,you must be in Decimals of a Degree. Before calculators, this had to be done by hand using math Base ( 60) Sexagesimal System.This was some time referred as "Babylonian Sexagesimal Time System".
    A system with a base of 60 might appear cumbersome, but it has distinct advantages, as most modern users have come to recognize, especially with respect to the number of integer divisors and associated reciprocals, i.e., 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 (base-10 has only the related pair 2 and 5). Convenient divisors are not limited to integers alone, but although the equivalents of recurring decimal fractions such as 0.333* and 0.666* are simply 0;20 and 0;40, awkward recurring sexagesimal numbers also exist, e.g., the fractions: 1/7 = 0;08,34,17,08,34,17,08,34,.. and 5760/177 = 32;32,32,32,32,.., the former perhaps influencing the choice of 3 (or 25/8) rather than the fraction 22/7 for PI. On a more practical note, the accurate (but still rounded?) Babylonian mean synodic month of 29;31,50,8,20 days (29.530594...days; modern value: 29.5305903...days) provided a fundamental unit of time and motion in Seleucid Era astronomy which could be replaced by an equivalent interval of 30 "mean lunar days" (tithis) to simplify computation. From a modern viewpoint the duration of one "tithi" would be obtained from the mindless division of 29;31,50,8,20 days by 30. However, instead of starting with "29" and carrying out successive divisions in the modern manner, all that is necessary is to start at the other end, multiply by 2, and shift the sexagesimal place accordingly to obtain the required result of 0;59,03,40,16,40 tithi per day. This is a very trivial example; for a vastly more complex application ."On the Big 6-Place Table of Reciprocals and Squares from Seleucid Babylon and Uruk, and their Old Babylonian and Sumerian Predecessors. It is hard to conceive that some how they had worked out all this. We owe all of this to them. This is one of the Stories of Math,Have you every seen this Documentary. The story of Math, I recommend it. Now sexagesimal, (redirected from Hexagesimal), Related to Hexagesimal: Hexidecimal , these three are all related to each other, sexagesimal and math base 60, I have used the most being in reference too the math I used to do this. Pertaining to a multiplicity of 60 distinct alternative states or conditions or, simply, a positional numeration system to radix (or base) 60. Or simply DMS to Decimal,and Decimal to DMS,using HMS or DMS they are the Same. How this all came about and they solve this system,is one of the math mysteries, But here is how we done it before calculators could. and it is not that hard.
    Lets take the angles of a Right Triangle of the multiples of 3,4,5, this is well known on how to square to lines at 90 Degrees, any multiples of 3,4,5 or those numbers when laid out will form 90 degrees,if your line is 3ft and 4ft the diagonal line from there ends will equal 5 ft. as in 2x3=6,2x4=8,2x5=10,and you will have the same triangle 2x bigger and this goes on3x3=9,3x4=12,3x5=15,this is the Right Triangle of multiples of 3,4,5, every builder and carpenter knows this equation, there are three Angles 90, 36.52-12 DMS and 53.07-48 DMS these two angles are well known,there placement depends on where the X and the Y is placed the Hypotenuse is always 5,but to do any math other than just simply adding or subtracting in base 60,you must be in decimals.even adding or subtracting them is easier in decimal, You do not have to change the Degree hold number so your math is only in minutes and seconds, or in decimals fractions.So Once Upon a Time we done it like this
    36.52-12 ,so -36 =.52-12 ,so 12sec /3600=0.0033,52 min /60=0.8667 (0.0033 +0.8667 )= 0.8700 or .8699 so now you have 36.8699 or round to 0.8700,now lets take it back 36.8699° -36 =0.8699 so .8699 x 60 = 52.1940 -0.1940 = 52 min ,now 0.1940 x 60=11.64, so you have 36°52' 11.64" which can be rounded to 36.52-12. Now the other angle 53°07"48" or 53.13 or 53.1301 as it is well known. Again ,.07-48, 48 sec / 3600 =0.0133 ,07 min /60 =0.1167, ( 0.0133+0.1167) =0.13 or 0.1301 or 53.1301,
    Now 53.1301 -53 =0.1301, so .13 x 60 =7.8000 -7 =.8000 x 60 =48, so 53°07'48",using .1301 you get.13 x 60 =7.8000 -7 =(.8000) +(.0001 x 60)=0.0060 =.8060 x 60 = 48.36 .53.07-48.36
    These are just two angles,so in your everyday calculation,how much math would you do ? Being said you may only convert one way DMS to Decimals ,but still that is a lot of math,so when you tap on your data collectors screen, Remember Once Upon a Time this is how a surveyor done it,and think again about the math that you did not have to do.

  • Hi Everyone,Once Upon a Time, Surveyors where more in touch with there work,mathematically ,because they had to do the math,this was not a push button world.One of the best improvement,to how they worked ,was a calculator.before,the inventions of science calculators, I did have a electronic,hand held calculator,and even though,it was not scientific,it did help with basic math. In 1972 hp came out with the first hand held scientific calculator.In 1960 there were scientific calculator,but no hand held,not till 1972,with Hp 35,it only had 35 keys Introduced at US $395. There were program calculators,more like early computers by 1960,these worked off CORDIC programming, CORDIC was conceived in 1956 by Jack E. Volder who worked for Convair that was into aeroelectronics because they build aircraft,by 1968 Hp release the HP 9100 a desk top calculator, that was program thru the Wang LOCI-2,it was one of the first uses of the CORDIC algorithm for trigonometric computation in a personal computing device, as well as the first calculator based on Reverse Polish Notation (RPN) entry. this is what set Hp apart from every one else. Texas Instruments (TI), after the introduction of several units with scientific notation, came out with a handheld scientific calculator on January 15, 1974, in the form of the SR-50. TI continues to be a major player in the calculator market, with their long-running TI-30 series being one of the most widely used scientific calculators in classrooms.But before all of this Slide Rules where used to do all your trigonometric calculations,this included,combined multiplication and division,proportions.circular measurements ,square roots,and squares,cube roots and cubes,logarithms,complex numbers and vectors,Base E, and Base,powers of E,powers of Base ,roots and
    common fractional Exponent,solving Exponential equations and logarithms of complex numbers. This is quite a list of what a side rule would do.Mine is a Log Log, this is a logarithm slid rule. So one took care of this slide rule.One of the first things that a calculator done was make all this faster,So now a person in the field could apply this on the job, in a reasonably
    time frame. This help Surveyors how to solve the math problems of horizontal and Vertical curves,So once Upon a Time, we done it like this.and i will use some old equations and methods,so Here is Horizontal Curves ,In once Upon a Time.
    1.The Degree of a curve and it Radius,
    D= 5729.58 / Radius , R = 5729.58 / Degree,,so for a curve of D=1°The Radius is 5729.58,notice that R is exactly inversely proportional to D.So a curve of 25.40-20" Degrees, the radius would be 223.181 'ft.
    A 5 ° curve the Radius would be 1,145.9160 ' ft. D =18,000 ÷ ( pi x R )

    or R = 18,000 ÷ ( pi x D ) if one is known,the other can be solve.
    2.Delta Angle ,most of the time this was taken in the field,before it was solved.

    Used for location only,if needed.
    I or Delta angle(DA) = Length / 100 x Degree o. c. = Intersection angle or Vertex, angle but lets define them.I is the intersection deflection angle of the two tangents.the Vertex,the point of intersection at the PI,so a point.where this takes place.I is also equal to the angle between the two Radii,that forms a triangle,with these two Radii and the Long Chord. solving this triangle will also check your chord distance.Next you need the two tangents of the Curve.
    I will refer to them as TL Tangent Length as not to be TAN
    3.TL = Radius x Tan of 1/2 the Delta angle or TL=R Tan(DA/2) or 1/2 Delta.
    4.C =Chord or long Chord , Is a Straight line distance , PC to PT,This is the bottom of the Segment,the area from the chord to the ARC .Long Chord or LC= (2 x TL) x Cos 1/2 DA or C=(2 x TL) x COS(DA/2)     or 2 x Radius x SIN (DA/2) or half Delta Angle.
    5.Curve length or CL or L, this is the distance in the ARC from the PC to PT.                          CL = pi x R x DA / 180 or CL =pi x R x DA ÷ 180
    L = R x DA ,where DA is in radians
    6.External Distance = E. This is the distance from the Vertex to the middle point of the Curve.E , can equal, R = E x COS( DA ÷2) ÷ ( 1 - Cos (DA ÷ 2)) solving E, but this is a Radius equation, it can be done.Here is a little more friendlier E =TL x TAN (DA÷ 4) or E = TL x TAN x 1/4 DA,here is some more, E = R ÷ COS ( DA ÷2) - R . or E = R exsec( DA /2 ).
    7. Middle Ordinate or MO or M, is the distance from the MID- Point of the curve to the MID -point of the Long Chord .
    M = The same here this is solving the Radius, but this can be done.                                        M= R = M ÷ (1 - COS ( DA ÷ 2 ) or M = E x COS ( DA ÷ 2 )
    Here is more, M = R x ( 1 - COS ( DA ÷ 2 )) or M = R vers( DA ÷ 2 )
    So there you have it,every thing you need to calculate a H Curve.I have listed all the Parts of a Curve 1. I angle or Delta Angle,2. Radius, 3.Degree, 4.Tangent Length 5. ARC Length ,6. Long Chord, 7.External ord.,8. Middle ord., Now it's fiscal parts 9. PC, Point Of Curve,10. PI, Point Of Intersection of the Vertex,11. PT , End of Curve,and beginning of a Tangent,as
    Point Of Tangent, This is a Simple Curve,but this how it was done,there were no programs,so think about this when you turn your Data Collector ON, But what happens when it does not turn ON,  Murphy's Law, will catch up with you.
    This does not show you how hard it was to convert HMS to Decimal, and what is next ,that and how to solve Deflections Angles,to lay out a curve. some of the first scientific calculators ,did not have HMS keys to do base 60 math,but this does not mean it was easier,you can do that on any basic math calculator. But stay tuned for Once Upon a Time. And how to lay out a Curve, Just you and your Calculator and your TS.and some Deflection Angles.

    • thanks for sharing I am proud to hold my hp50g

  • Party Chief

    Wow Billy.. just wow.. need to read this again a few times.. thanks for sharing..

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