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**Introduction to the "Triangle Series"**

A cacher who is new to some of these concepts is strongly advised to do these caches in alphabetical order: Archimedes (GC3TT8C), Bernoulli (GC3V9MW), Cayley (GC3W898), Descartes (GC3X0W7), Euclid (GC3XGN0), Fermat (GC413V5), Gauss (GC4219Z), Heron (GC4VGBX), Ingham (GC4WC72), Jarnik (GC4Z9PK), Kepler (GC4ZMRG), Lovelace (GC50BN6), Merryman (GC51D34) and Newton (GC5AJ04).

**Opal & Yugsirap strike again!! Congratulations on yet another FTF feather for your bonnet!**

**Introduction to this Cache**

**Newton is the FINAL cache** in this series having to do with triangles.

**Newton, the Person (1643 - 1727)**

In 1661, after having shown little promise during an unhappy childhood, Isaac Newton entered Trinity College, Cambridge, to study law. He became interested in astronomy through exposure to scientists of the past but, being lost in the mathematics involved, he turned in that direction starting with a study of Euclid’s deductive geometry. He received his bachelor’s degree in 1665.

With the onset of the “Great Plague” in 1665, the university closed so Newton went home and continued his personal studies there. During this two-year period, he established the basics of differential and integral calculus. Returning to Cambridge in 1667, he was elected to a “minor fellowship” followed quickly by his earning his master’s degree and being elected to a “major fellowship” in 1668. A year later, he was appointed as the “Lucasian professor of mathematics,” a position he held until he left the university thirty-three years later.

During this period he studied optics and reached the conclusion that instead of white light’s being a single entity, it is, as we know today, a mixture of all the colours of the spectrum. He constructed a reflecting telescope in an attempt to avoid the breaking up of white light by the lenses in the telescopes of the day. He was made a Fellow of the Royal Society in 1672. Continual disagreements with his peers over his opinions on the nature of light contributed to a “nervous breakdown” in 1678.

Probably Newton’s greatest contribution was in the area of celestial mechanics. His work here led to the “law of universal gravitation” which states that there is a force of attraction between any two objects in the universe which acts along a line between their centres of mass and is proportional to the product of the masses of the two objects and inversely proportional to the square of the distance between those centres. In other words, the more massive the objects, the bigger the force of attraction and, similarly, the closer they are the bigger the force. In 1687, he published what is arguably one of the greatest scientific books ever written, commonly known as *Principia* - a full treatment of his physics theories and their application to astronomy. By then, he was established as an international leader in scientific research.

A second “nervous breakdown” in 1693 led to his retirement from research and he turned his interest to government positions in London although he did not retire from Cambridge until 1701. He was elected president of the Royal Society in 1703, a position he held until his death, and was knighted in 1705 - the first person to achieve this status because of his scientific achievements.

**Newton the Cache**

**Background: Radian Measure and Your Scientific Calculator in the Field**

The completion of Merryman is essential to this cache. The only new concept being introduced is that of "radian measure." The work on the puzzles will depend on that and the processes used in Merryman.

**Radian Measure**

Draw a neatish circle of approximately 5 cm radius. Draw a radius therein - a line from the centre (O) meeting the circumference at P. Now estimate from P an arc along the circumference equal in length to the radius ending at Q. Now join OQ thereby adding a second radius to the diagram. The angle at the centre of the circle between the two radii is 1 radian. Formally, **a radian is the angle subtended at the centre of a circle by an arc along the circumference equal in length to the radius of the circle.**

**How Do Radians Compare to Degrees?**

Looking at the same diagram again, if you were to join P and Q with a straight line, that straight line - representing the shortest distance between P and Q - would complete a triangle which is almost an equilateral triangle - but not quite - because that third side is slightly shorter than the radius - i.e. the other two sides of the triangle - because the arc of the circle (which is longer than that straight side) is equal to the radius. If the triangle were equilateral, each angle would have 60° Since that side at the circumference is slightly less than the other two sides which are radii, the angle at the centre - that 1 radian - would be less than 60°. In fact, it is approximately 57.3°. How do we know that? Bear with us!

**The Relationship Between Radians and Degrees**

It is commonly known that the circumference (C) of a circle is calculated by a formula using the radius and the constant "pi" (a Greek letter) which is represented by π. (It is supposed to look somewhat like an eleven with a squiggly line across the top extending out a little on each side but for some reason it looks like a large lower-case "n" in this document.) The formula is:

C = 2 πr - meaning that to calculate the circumference one multiplies 2, π and the radius all together. If you divide both sides of this little formula by the radius, r, you get:

**C/r = 2 π**.

How would you find how many 4 m cedar rails there are along the top of a **straight** cedar fence of 276 m (assuming no overlap, of course)? You would divide the entire length of 276 by 4, resulting an answer of 69. How would you find how many radii lie in the circumference of a circle? As in the fence example, you would divide the entire length, C, by the smaller length, r.

And what do you get when you divide C by r? We have just seen that **C/r = 2 π** meaning that the circumference is made up of 2 π radii. Since one radius of arc forms an angle of 1 radian at the centre of the circle, then 2 π radii form an angle of 2 π radians at the centre of the circle.

But we know that there are 360° in a circle. So, an angle of 2 π radians is the same as an angle of 360°.

So 2 π radians = 360°.

Dividing each side of that statement by 2, we arrive at the generally used equivalent:

**π radians = 180°**

Dividing each side by π yields 1 radian = 180/π°.

**But How Do We Work with “π?”**

Many will remember - perhaps vaguely! - using the fraction 22/7 or, perhaps, the decimal 3.1416 as approximations for π. But they are just that - approximations. With **scientific calculators, which are essential for this cache**, we no longer have to do that. There is a button on the calculator - on the button itself or on the body of the calculator (which dictates the use of the shift/second function button) which will give us the exact value of π any time we want to use it. So we never write down a number for π; we just use the symbol in writing and activate the appropriate calculator button when using it in calculations.

So, given that 1 radian = 180/π°, using a calculator to evaluate that fraction gives you a degree value for one radian of approximately 57.296 - or 57.3 as mentioned above.

Now we hope that you have understood all that and that you appreciate being reminded of it but none of it is necessary to complete the work for this cache!! (**Now** you tell us!!)

**So What are We Doing in this Puzzle Cache Anyway?**

**Beware!!! Some calculators default to “DEG” without warning so do ensure that your calculator is in the “RAD” mode for all the calculations in this puzzle!** (Not strictly necessary but we'll do it this way to prevent confusion!)

**The Puzzle!**

- Go to the stated co-ordinates.

**- Be sure to review “Merryman” (GC51D34) and to take a scientific calculator with you - be it in your GPSr or otherwise.** The given co-ordinates will take you to the first stage of a four-stage field puzzle multi. The solution of the field puzzle at stage one will take you to stage two and so on. Don’t round off!!

- The first three stages consist of micros - varying sizes of pill containers - all below knee height.

- The final is a well-cleaned peanut butter container just about at knee height.

- All co-ordinates were based on an average of 200 trials minimum and were identical on more than one day.

- The outstanding feature of this trail is the sky - enjoy it!

**Good luck!!**

**Optional - but Important - Supplement: Finding the Given Stage One**

**Introduction**

You have the co-ordinates of Stage One already - the stated co-ordinates of the cache. This section, by which you can find the co-ordinates of Stage One and thereby confirm that you know what you are doing, is to demonstrate the sorts of calculations you will be expected to carry out in the field to help you move on through the four stages. You might want to do it to ensure that you get the co-ordinates given at the top of the cache page - i.e. that you have the right idea of what to do out there. Then, if you’re having a problem, you can contact someone - even the CO - to find out where you are going wrong. It would be too bad to find yourself at Stage One but unable to get yourself to Stage Two!

**The Sort of Thing You Will Find at Each of Stages One, Two and Three**

To get to Stage One, you are looking for N 44̊ 19.ABC’ and W 078 40.DEF.

A is the hundredths digit of the sine of 0.28 radians. ___

B is the thousandths digit of the cosine of the angle of which the sine is 0.9769. ___

C is the ones digit of the secant of 5.01 radians. ___

D is the thousandths digit of the tangent of the angle of which the cosecant is 1.850. ___

E is the tenths digit of the angle in radians of which the cotangent is 0.2035. ___

F is the thousandths digit of the cosecant of the angle of which the cotangent is 0.2574. ___

**
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