Simulators

One of the problems with learning to use an astrolabe is that in order to work through the class notes the student needs to first print out and assemble their own astrolabe from the provided materials. This takes time and materials that may not be ready to hand.

What I’ve wanted to do for a while now, was to provide an online simulator to allow the student to work without a physical copy of the device. In the past I ave played with Flash and Java, but these had significant problems with getting them to work smoothly with the Internet. Instead, I settled on HTML5 and JavaScript. Much more flexible and simple to use, and supported by all modern browsers.Simulator

There are currently two simulators available: One for the astrolabe and one for the sine quadrant. These are a work in progress and improvements are planned, but they work well as they are.

http://www.astrolabeproject.com/sim/astrolabe/sim.html
http://www.astrolabeproject.com/sim/sine/quad.html

The simulators have been tested on current versions of Internet Explorer, Firefox, Chrome, and Safari. (Note: if you are on an IPad, you can save the simulator page to your desktop and it will function like an app, still requiring an Internet connection though).

The Astrolabe Simulator
The astrolabe simulator can be found at (link). Clicking and dragging the rate rotates it, the same with the pointer arm. Clicking on the pushpin on the upper right locks the rete and arm together and allows them to be moved together. Clicking on the icon upper left flips the astrolabe to the back side.

The Sine Quadrant Simulator
The sine quadrant simulator can be found at (link). The cord can be rotated by clicking and dragging the weight on the end. The index bead (green circle) can be moved by dragging it. Function lines can be toggled on and off by clicking on the check marks bottom right.

The source code for both simulators is open source and can be downloaded here(link).

Enjoy!

Introducing the Advanced Astrolabe

For quite some time now I have been basing my class, and my discussions here, on an astrolabe with fairly basic options. Linked to the right you will find a zip file containing the components for this basic astrolabe, along with the handout for my class.

But there is more to the astrolabe than just the basic functions I have discussed so far. An examination of astrolabes from the medieval period shows a wide range of advanced function options available to the users. Over time, I have started to add some of these slowly to the Astrolabe Generator application, so that people out there can play with them. But up to now I have not spent any real time discussing of how they are used. Over the next few weeks I hope to examine several of these advanced functions.

I have rearranged the links on the left and added a new download: An Advanced Astrolabe example. This differs from the Basic example in several ways. On the front you will see new markings on the plate for the Houses of Heaven; on the back I have added a sine/cosine scale (upper left), an Arcs of the Signs scale (upper right) and the Lunar Mansions (center). In the next few posts I will discuss the purpose and use of each of these in turn.

 Advanced astrolabe

Advanced Astrolabe Functions: Estimating the direction to Mecca.

Given that the math and sciences of the Middle East were a major driving force in the development of the astrolabe[1], it is reasonable that some of the advanced functions are related to the dominant religion of the area, Islam. Today we will discuss using two of the astrolabe’s advanced features to estimate the direction of Mecca.

For a Muslim, knowing the direction to Mecca from one’s current position is not just important because of the requirement to pray in that direction five times daily. In addition, the ritual butchering of food animals needs to be done with that direction in mind for the meat to be acceptable to the dietary laws; and when bodies are buried, they need to be placed on their sides, facing toward Mecca[2]. The specific location of the Kabaa, the focus of prayer is 21.4N 39.8E[3].

Determining this direction, known as the Qibla, is a major subject in Islamic mathematics. The Qibla is defined as being the “Great Circle” direction between a given point and the location of Mecca[4]; and there are many treatises devoted to computing it accurately. But in the case of a traveler, without access to intricate mathematics or carefully computed tables, finding the proper direction is tricky.

There are two scales to be found on the back of a typical astrolabe of the region that can help with this problem.

Estimating the Qibla using the Sine/Cosine scale
It is possible to get a rough estimate of the qibla direction by using the Sine/Cosine scale to be found on the back of many astrolabes[5]. To use this scale the user must know the latitude and longitude of Mecca, as well as their own current latitude and longitude[6].

To use the sine/cosine scale, work the problem as follows:

  • First, find the difference in latitude between the two positions
  • Next, find the difference in longitude.
  • Set the pointer or alidade of the astrolabe to point at an angle from horizontal equal to the difference in latitude. Mark the line from there to the sine scale mentally.
  • Set the pointer or alidade to point at an angle from vertical equal to the difference in longitude. Mark the line from there to the sine scale mentally.
  • Set the pointer to the point where the two lines cross. The pointer will now point to a direction. You will need to determine whether the direction given is to the North, South, East, or West. For example, if you are North and East of Mecca, the qibla direction will be measured from West.

Finding the direction to Mecca using the sine/cosine scale

Example:
Finding the Qibla from Alexandria Egypt (location: 31.2N 30.0E)

The difference in latitude (rounding off) is: 10 degrees
The difference in longitude is: 10 degrees

Work the problem as below: Measure up ten degrees from the horizontal scale(1) for the difference in latitude(2), and ten degrees from the vertical scale(3) for the difference in longitude(4). Place the rotating arm on the intesection of the two imaginary lines(5) and it will point at an angle(6), 45 degrees in this case. As Mecca is east and south of Alexandria, this angle will be the angle south of due East. So add it to 90 degrees to get the compass bearing (135 degrees). Using the ruler function in Google Earth (which conveniently uses great circles) The great circle direction to Mecca is 135.68 degrees.

Finding the direction to Mecca using the sine/cosine scale

Determining the Qibla using the Qibla line on the arcs of the signs scale
On many astrolabes there is a scale that shows the altitude of the Sun at Noon for every day of the year. This scale is in the form of a series of ninety-degree arcs marked with the various zodiac symbols representing the location of the sun on the ecliptic. In addition to one or more curved lines representing the altitude of the Sun at noon, many of these are also marked with a set of arcs labels as major cities. These arcs can be used to determine the direction of the Qibla if the user is near one of the cities listed.

To determine the rough Qibla direction using the arcs of the signs, do as follows:
– find the Sun’s position on the zodiac using the zodiac and calendar rings of the astrolabe.
– Find the city line for the city you are near.
– Find the point on that line that is in the arc for the sun’s current zodiac position.
– Set the astrolabe’s alidade/pointer to touch this point. The alidade will point to an angle on the rim.
– When the sun is at this angle above the horizon, it will sit directly above the Qibla direction.

Finding the direction to Mecca using the arcs of the signs

Example: Find the direction to Mecca from near Bagdhad Given the date is the Spring Equinox (March 20th).

  • First find the position of the Sun on the ecliptic for the Spring Equinox (I cheated to simplify. The First Point of Aries marks the Spring Equinox, no need to look it up).
  • Rotate the alidade/pointer to the point where the Bagdhad line crosses the arc marking Aries 0 (1) and it will point to an angle(2). In this case, for this date, 55 degrees. So when the sun is at an angle of 55 degrees, it is shining directly over the point on the horizon that marks the direction to Mecca.

Limitations
Clever as these techniques are, they are limited in usefulness. As the Qibla is a great circle line between points on the globe, the further you are from Mecca, the more inaccurate the estimate found by the first method will be. For example in the city of Casablanca (location:33.5N 7.5W) the Qibla direction is about 93.5 degrees but if you estimate it using the sine/cosine scale you get a direction of 106 degrees.

The second, arcs of signs, method is similarly limited. It does provide the great-circle direction, but because the sun has to be up and in the direction of Mecca to work, it is pretty much limited to locations within 90 degrees of Mecca.

Notes:
[1]Morrison, James E., The Astrolabe, Janus, 2007, pg. 31
[2]King, David A. and Lorch, Richard P., “Qibla Charts, Qibla Maps, and Related Instruments”. The History of Cartography, Volume 2, Book 2: Cartography in the Traditional East and Southeast Asian Societies, University Of Chicago Press, 1995. pg 198.
[3]http://en.wikipedia.org/wiki/Kaaba
[4]http://en.wikipedia.org/wiki/Qibla
[5]This technique also works on the sine quadrant, of course.
[6]King, David A., World Maps for Finding the Direction and Distance to Mecca: Innovation and Tradition in Islamic Science, Brill:1999, pp 57-60

Deconstructing the Sine Quadrant-Part 6: Summary and sources

Links

The previous installments in this series can be found here:
Part One
Part Two
Part Three
Part Four
Part Five
Printable example Sine Quadrant

In the last 5 installments we have broken down the various parts and functions of the sine quadrant. I’ve now come to the end of what I currently know. So here are some references if you wish to dig deeper.

But first some questions I still have:

  1. Why do some examples of the sine quadrant have the Asr arcs reversed? On some examples I have seen (see below), the arcs extend from the 90 degree point down toward the cosign (vertical axis) not from 0 toward the horizontal sine axis.
  2. Why do many examples of the sine quadrant contain both a 12 unit and 7 unit Asr line? These give the same answer, there is no reason I can see for marking both. The traditional height of a gnomon in the Middle East is 7 feet, and the shadow square of an astrolabe often has a 7 unit side for that reason. But why 12 units then? Perhaps just because 12 divides up easily into halves, thirds and quarters?
  3. Why are the Asr lines always drawn on both the horizontal and vertical?
  4. What is the purpose of the line that is often drawn as a chord from 0 to 90? on several examples below the line is labeled, indicating some use, but what? I can find no reference.
  5. I can see why the angle scale is often labeled in both directions (to allow measuring distance from zenith); but why are the sine/cosine scales also marked in both directions?

Extant Sine Quadrants Online
http://en.wikipedia.org/wiki/Sine_quadrant
This is the Wikipedia entry for the Sine Quadrant. Not much information is available concerning the example in the pictures, but note that the Asr arc appears to be a later, and crudely done addition.

http://brunelleschi.imss.fi.it/galileopalazzostrozzi/object/TobiasVolckmerQuadrant.html
— This is a European variant. Note that this one has scales divided into 90 units as opposed to the usual 60. Note also the vernier scale on the degree arc. This allows for increased precision in measuring angles

http://iin-aina.blogspot.com/2011/04/nemu-kuadran-terbesar.html
— A quadrant of unusual size…

http://blogs.unpad.ac.id/kkltaringgultengah/sosial-budaya-desa-taringgul-tengah/
http://blogs.unpad.ac.id/kkltaringgultengah/files/2012/02/oo-1024×681.jpg
— A nice example of a wooden sine quadrant, the scales outside the degree scale show the zodiac and are for determining the position of the sun for finding the Solar declination.

http://www.mhs.ox.ac.uk/collections/search/displayrecord/?mode=displaymixed&module=ecatalogue&invnumber=48133&irn=1331&query=
http://www.mhs.ox.ac.uk/collections/imu-search-page/record-details/?TitInventoryNo=16856&querytype=field&thumbnails=on&irn=8355
http://www.mhs.ox.ac.uk/collections/imu-search-page/record-details/?TitInventoryNo=36052&querytype=field&thumbnails=on&irn=3334
http://www.mhs.ox.ac.uk/collections/imu-search-page/record-details/?thumbnails=on&irn=8351&TitInventoryNo=15598
http://www.mhs.ox.ac.uk/collections/imu-search-page/record-details/?thumbnails=on&irn=7066&TitInventoryNo=42166
http://www.mhs.ox.ac.uk/collections/imu-search-page/record-details/?thumbnails=on&irn=7083&TitInventoryNo=35285
http://www.mhs.ox.ac.uk/collections/imu-search-page/record-details/?thumbnails=on&irn=7465&TitInventoryNo=28781
http://www.mhs.ox.ac.uk/collections/imu-search-page/record-details/?thumbnails=on&irn=3343&TitInventoryNo=35612
http://www.mhs.ox.ac.uk/collections/imu-search-page/record-details/?thumbnails=on&irn=8350&TitInventoryNo=32534
http://www.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectId=178957&partId=1&searchText=quadrant&images=true&page=2

http://www.mhs.ox.ac.uk/epact/catalogue.php?ENumber=35728&Level=
http://www.mhs.ox.ac.uk/epact/catalogue.php?ENumber=96841&Level=
http://www.mhs.ox.ac.uk/epact/catalogue.php?ENumber=94421&Level=
http://www.mhs.ox.ac.uk/scienceislam/objects.php?invnum=35285

http://collections.rmg.co.uk/collections/objects/43268.html
http://collections.rmg.co.uk/collections/objects/43269.html
http://collections.rmg.co.uk/collections/objects/43256.html

References

Bir, Atilla. (2008). Principle and Use of Ottoman Sundials. Retrieved from http://www.muslimheritage.com/topics/default.cfm?ArticleID=942

Charette, François, Mathematical Instrumentation in Fourteenth-Century Egypt and Syria. The Illustrated Treatise of Najm al-Din al-Misri, Brill, Leiden (2003).

Morley, William H., Description of an Arabic Quadrant, Journal of the Royal Asiatic Society of Great Britain and Ireland, Vol 17 (1860), pp. 322-330

 

Deconstructing the Sine Quadrant-Part 5: The specialized lines – Continued

In part four we examined the Asr lines and how to use them to find the proper times for the start and end of the Asr prayer required of all Moslems. This week we are going to look at an alternate method of determining that by examining yet another of the sine quadrant’s advanced functions.

Links
The previous installments in this series can be found here:
Part One
Part Two
Part Three
Part Four
Printable example Sine Quadrant

 Sine Quadrant

The Other Asr Lines
The two arcs described in the last part of this series are one method of determining the times for the midafternoon Asr prayer, but there are alternative methods. Some sine quadrants do not have the arcs, but rely on a different set of markings. Depending on the maker, devices will have one set of markings or the other, or both, or none.

If you look at the example sine quadrant pictured above, you will see, in addition to the previously discussed Asr arcs, two lines of markings (marked in red dots) parallel to the horizontal and vertical scales at 7 and 12 units respectively.

asr_lines.png

Traditionally the markings at 12 units are most often seen, but I have seen several examples with the additional markings at 7 units. It does not matter which set are used, as the answer they give is the same. For clarity, I will concentrate on the 12 unit line, but the steps I describe work just as well for the 7 unit line.

As was discussed in part four, the time for the start and end of the Asr prayer are defined by the length of a shadow: The period for Asr begins when the shadow of a vertical pole is equal to its noontime length plus the length of the pole; and ends when the shadow is the noontime length plus twice the length of the pole[1].

The vertical line at 12 units can be used to simulate a pole 12 units high[2]. By marking where the cord crosses the line when it is set to the Sun’s noon angle you know the length of the shadow at noon. Then by adding 12 units and moving the cord to cross at that point, you can compute the angle of the Sun at the start of Asr.

To demonstrate that using the Asr arc and the Asr line both give the same answer, let’s work through an example where the Sun is at a height of 50 degrees at local noon.

First set the cord of the quadrant on the 50 degree mark (A). Note where it crosses the Asr arc at (B), and follow that line down to find an angle of 28.5 degrees. Now notice where the cord crosses the 12 unit Asr line at 10 units (C). add 12 units to this (the “height” of the “pole”), to get 22 (D) and move the cord to cross at this point. The cord will lie at the expected angle, 28.5 degrees (E).

 Finding Asr on the sine quadrant

To find the angle of the Sun at the end of Asr, you just need to add an additional 12 units to simulate adding twice the length of the pole.

Notes:
[1]Bir, Atilla. (2008). Principle and Use of Ottoman Sundials. Retrieved from http://www.muslimheritage.com/topics/default.cfm?ArticleID=942

[2]Charette, François, Mathematical Instrumentation in Fourteenth-Century Egypt and Syria. The Illustrated Treatise of Najm al-Din al-Misri, Brill, Leiden (2003). Pg 176-177

 

Another astrolabe in work

This one is a metal astrolabe. A work in progress that uses some of the output of the Astrolabe Generator.

Astrolabe in progress

Astrolabe in progress

Images used with permission.

Deconstructing the Sine Quadrant-Part 4: The specialized lines – Continued

In Part Three we examined the obliqity arc and how to use it to find the local angle of the Sun at Noon. This week we are going to look at one use for that information, and in doing so, examine another of the sine quadrant’s advanced functions.

Links
The previous installments in this series can be found here:
Part One
Part Two
Part Three
Printable example Sine Quadrant

Sine Quadrant

The Asr Lines
One of the major uses of the sine quadrant was computing the proper times for the Muslum mid-afternoon prayer, Asr[1]. To make this task easier, there are often special lines cut into the face of the quadrant.

Prayer Times
Traditionally, the times set for the 5 daily prayers required by the Muslim religion are based on the Sun’s position in the sky[2]. Of interest to us today is the mid-afternoon prayer, Asr. The period for Asr begins when the shadow of a vertical pole is equal to its noontime length plus the length of the pole; and ends when the shadow is the noontime length plus twice the length of the pole.[3]

Shadows.png

Because Asr is based on the Sun’s shadow, the times will vary depending on location, and determining the proper times for prayer must be done for each day. The sine quadrant allows the user to determine these times accurately and quickly.

If you look at the example quadrant, you will see a pair of shallow (almost straight) diagonal curves, marked in red. The curves run from the zero degree mark up to roughly the middle of the horizontal scale. The lower line is used to compute the start of Asr, the upper (often missing) allows you to compute the time for the end.

By the definition above, both times are based on the Sun’s position at Noon; This gives us one use for finding the Sun’s noontime altiude. Once the angle of the Sun at local Noon is found, it is just a matter of marking where that line crosses the two Asr curves, to determine the Sun’s angle at the beginning and end of Asr.

For example: Let’s work through finding the times for Asr for 31 degrees North (just south of the city of Alexandria, Egypt), for the 15th of February. If needed, you can go back and review [part 3] concerning the use of the obliqity arc.

First, determine the Sun’s angle at Noon:
The 15th of February is 56 days after the Winter Solstice on December 21. So, remembering that the solstices are at the 90 degree mark of the angle scale, we count down 56 degrees and place the cord at 34 degrees (A). Make a note of where the cord crosses the obliqity arc (B), and follow that point down to get an angle of 13 degrees(C).

Finding noon with a sine quadrant

Remember that this is Winter, so the Sun is in the South and therefore the declination is negative, giving us a solar declination of -13 degrees. Using that and our known latitude we compute the Sun’s angle at noon to be:

(90-31) + -13 = 46

Now that we know what the Sun’s angle at Noon will be, we can determine its angle for the beginning and end of Asr very easily using the Asr lines: Place the cord on the computed Noon angle(A), and note where the cord crosses the first Asr line(B); follow this point down to the angle scale to find that the angle of the sun at the start of Asr is 26.75 degrees(C). Now note where the cord crosses the second Asr line(D), and do the same to find the angle of the Sun at the end of Asr to be 18.5 degrees(E).

Finding asr with a sine quadrant

After these two angles are known, it is just a matter of checking the Sun’s altitude at regular intervals to know when it is time for the prayer.

In part 5, we will examine another method of determining Asr.

Footnotes:
[1]Bir, Atilla. (2008). Principle and Use of Ottoman Sundials. Retrieved from http://www.muslimheritage.com/topics/default.cfm?ArticleID=942

[2]King, David A. A Survey of Medieval Islamic Shadow Schemes for Simple Time-Reckoning. Oriens, 32(1990), 196-197. http://www.jstor.org/stable/1580631

[3]Note: There are various schools of thought, and regional and cultural variations. The above is not definitive and is based on several sources.

An Astrolabe in Work

I got this last night via Twitter:

An astrolabe being assembled

https://twitter.com/cforchino/status/434176137216794626

Nice work, he has a steadier hand than I do. Please do email or tweet me your astrolabes, I’ll be glad to post them. Addresses are on the right.

 

Deconstructing the Sine Quadrant-Part 3: The Advanced Functions Continued

In part two we began examining the functionality provided by the various lines and arcs visible on the front of a typical sine quadrant. The use of the sine and cosine arcs was fairly straight-forward, being related to the basic function of the device (converting from angle to sine/cosine and back). Next we are going to examine one of the more complex functions; one that will be very useful in later installments.

Links
The previous installments in this series can be found here:
Part One
Part Two
Printable PDF of the example Sine Quadrant

 


Sine_Quadrant.PNG

The Obliquity Arc
Another line found on many examples of the Sine Quadrant is a circular arc centered on the origin point at the quadrant’s right angle. With a radius of approximately 24 units (marked in black on the figure above) this arc is a projection of the Earth’s orbital obliquity (tilt). The purpose of this marking is to allow the user determine the Sun’s declination (angle above or below the equator) for any given day, allowing the user to then determine the sun’s altitude at noon for that day(besides being neat, this will be of use later on: See part four.)

To understand how this works we will need to review a bit of basic astronomy:

The Earth’s axis of spin, and therefore its equator, is tipped 23.4 degrees to the plane of the planet’s orbit.

This means that as the Earth moves around the Sun in the course of a year, the Sun appears to move back and forth over the equator spending time in both the Northern and Southern hemispheres.

Traditionally, the Sun’s path through the sky (the ecliptic) is divided up into 12 30-degree zodiac signs. The Spring Equinox defines the zodiac’s starting point, the “First Point of Aries” (Aries 0), when the sun is directly over the equator getting ready to head north. At this point the day and night are of equal length. Each day the Sun moves a little way along the ecliptic, progressing through the zodiac signs until it returns to its starting place. As the days pass the Sun appears to creep north until the Summer Solstice, when the Sun is at its northern-most point and the day is at its longest. Then the Sun moves back south. The movement of the Sun and the seasons should be familiar to all.

Now look at the following diagram:

At the equinox the sun appears directly above the equator, so for someone standing on the equator, at noon the sun would be directly overhead, at an angle of 90 degrees to the horizon. If another person was standing at the North Pole at the same time, they would see the Sun on the horizon, or at 0 degrees elevation. Therefore, you can compute the angle of the Sun above the horizon for the equinox as:

noonAngle = 90 – lat

When the Sun is at a different part of the ecliptic, it will be up to several degrees north or south of the equator, so the equation becomes:

noonAngle = (90 – lat) + DecSun

Or to put it another way, you can compute the Sun’s noon altitude if you subtract your latitude from 90 and then add the Sun’s declination for that day.

Example: Given you are at 50 degrees north latitude, find the Sun’s noon altitude for the Summer and Winter solstices:

Summer: (90-50) + 23.4 = 63.4 degrees
Winter: (90-50) + -23.4 = 16.6 degrees (remember the Sun has a negative declination in winter/south of the equator)

Returning to our sine quadrant now: How can we determine the Sun’s declination for a given day?

The zodiac is divided up into 360 degrees. Think about it this way: At the Spring Equinox the Sun is 0 degrees on the zodiac; at the Summer Solstice the Sun is at 90 degrees; at the Fall Equinox, 180 degrees; Winter Solstice sees the Sun reach 270 degrees and finally returns to 0 at Spring Equinox again.

Now take the sine quadrant and hold the cord at 0 degrees. Note where it crosses the obliquity line and follow the grid down and read the angle: 0 degrees. Now move the cord to 90 degrees and do the same: you will get an angle of about 23.5 degrees.

Finding the declination

 

At this point reverse the direction you move the cord and move an additional 90 degrees to 180 (the cord is now back at 0). By moving the cord up and back four times you sweep out 360 degrees, and can simulate traversing the entire zodiac. Zero degrees represents the Spring and Fall equinoxes, and 90 degrees represents the Summer and Winter solstices.

So, let us say you want to know the angle of the sun at noon for the 5 of May. If you look up the Sun’s position for that date in your ephemeris (if you needed to use this device you would most likely have one hanging around), you get a figure of Taurus 15. Taurus is the 2nd symbol in the zodiac, so add 15 to 30; this gives us 45 degrees. Move the cord to the proper position, 45 degrees, and mark where it crosses the obliquity arc, follow this point down to the degree scale and read off the Sun’s declination as 16.5. You are still standing at 50 North (from the previous example), so you can compute the Sun’s elevation at noon for that day to be:

(90-50) + 16.5 = 56.5 degrees

So, what if you misplace your ephemeris? Then what? Well, there are 365.25 days in a year, and 360 degrees in a circle. Counting one day per degree only puts us off 5.25 degrees by the end of the year, this would translate to an error in declination of less than two degrees. If we are counting days since the last solstice or equinox, the possible error is only a quarter of that, probably within the observational error of the instrument. So, depending on how important fine accuracy is to you, you might not need an ephemeris at all.

Let’s rework the last example without an ephemeris:
The Spring Equinox is March 20th, therefore the 5 of May is 46 days later. So moving 46 degrees around from 0 (Spring Equinox) we place the cord at 46 and read a declination of 17 for the Sun. This is only slightly off (half a degree) from the figure we got above (16.5).

So. as we have seen, with a little mental calculation, a person can use a sine quadrant to find the angle of the Sun at noon for any day of the year. The obvious next question is why would you need to know? Next week I will be explaining that, and discussing some more of the functions of the sine quadrant.

 

Deconstructing the Sine Quadrant-Part 2: The Advanced Functions

In the previous post, I described the basic features of the Sine Quadrant, and described two of its most common functions: measuring angles and converting back and forth from angle to sine/cosine.

[If you need it, here is a link to the PDF of the example sine quadrant you can print out to follow along.]

In this and subsequent posts I will start to dig deeper, and examine the function of some of the other markings on the face of a typical sine quadrant.

The reader is now familiar with the sine/cosine grid and its uses, but a close examination of the face of our example sine quadrant shows several lines and arcs we have not yet discussed.

Sine_Quadrant.PNG

The Sine and Cosine Arcs
On many sine quadrants there will be two half-circle arcs (blue on the figure above), one centered on the Sine scale, one centered on the Cosine scale. These can be used in conjuction with the Sine and Cosine scales as an alternative method of converting angles to sine/cosine.

Note: If these arcs are to be used, there has to be a moveable bead on the weighted cord, this bead is used as a cursor to mark a position on the cord. Think of it as memory storage for the device.

To use these arcs, the procedure is similar to using the grid. First the user pulls the cord taught over the desired angle. Next, the user slides the index bead to rest directly on the appropriate arc (the horizontal arc for sine, the vertical for cosine). Once the marker is in place, the user will then rotate the cord to the sine or cosine scale and read the answer from the point under the bead.

In the figure below the cord is first set to 30 degrees(1), the bead is then positioned directly on the sine arc(2), then the cord is rotated to the horizontal sine scale(3) and the sine is read(4) as 30/60 or 0.5.

Converting to other way, from sin/cosine back to an angle is straightforward as well the user just lines the cord up with the scale, positions the bead at the given sine or cosine; then rotates the cord until the bead touches the appropriate arc. Then the cord will be set to the equivalent angle.