the others follow easily from these two. If both sides are multiplied by 2, these formulas are also called the Werner formulas. The algorithm Using the second formula above, the technique for multiplication of two numbers works as follows: Scale down: By shifting the decimal point to the left or right, scale both numbers to values between and , to be referred to as and . Inverse cosine: Using an inverse cosine table, find two angles and whose cosines are our two values. Sum and difference: Find the sum and difference of the two angles. Average the cosines: Find the cosines of the sum and difference angles using a cosine table and average them, giving (according to the second formula above) the product . Scale up: Shift the decimal place in the answer the combined number of places we have shifted the decimal in the first step for each input, but in the opposite direction. For example, say we want to multiply and . Following the steps: Scale down: Shift the decimal point three places to the left in each. We get and . Inverse cosine: is about 0.105, and is about . Sum and difference: , and . Average the cosines: is about . Scale up: For each of and we shifted the decimal point three places to the left, so in the answer we shift six places to the right. The result is . This is very close to the actual product, (a percent error of ≈0.8%). If we want the product of the cosines of the two initial values, which is useful in some of the astronomical calculations mentioned above, this is surprisingly even easier: only steps 3 and 4 above are necessary. To divide, we exploit the definition of the secant as the reciprocal of the cosine. To divide by , we scale the numbers to and . The cosine of is . Then use a table of secants to find out that is the secant of . This means that is the cosine of , and so we can multiply by using the above procedure. Average the cosine of the sum of the angles, , with the cosine of their difference, , Scaling up to locate the decimal point gives the approximate answer, . Algorithms using the other formulas are similar, but each using different tables (sine, inverse sine, cosine, and inverse cosine) in different