*Right-angle triangles with whole-number sides are called Pythagorean. The first is the triangle with sides of 3, 4 and 5. Eight of them form a square 7 by 7, leaving a central blank square 1 by 1.*Jonathan writes: Last week, John Michell told us why Geometry is not just some dull, dry topic but an ancient magical art. He spoke about the hidden patterns that link nature to music and architecture... and hinted at a further ‘cosmic connection’. This week, he goes a step further... revealing some of the ‘sacred ratios’ that may hold a clue to the meaning of life.

**THE SECRET NUMBERS AT THE HEART OF CREATION**

by John Michell

Cosmic geometry is very simple. You only have to know Pythagoras's theorem and a few tricks of the trade, and soon you will be able to construct the traditional plan of the universe.

Pythagoras was a Greek from Samos in the sixth century BC. He travelled to Egypt and other centres of learning, studying the science and philosophy behind ancient civilisations. He then expressed that knowledge in mathematical form, and taught it through music and geometry. From his famous theorem, we discover the other important ratios of ‘sacred’ or ‘creative’ geometry.

If the side of a square measures 1, the length of its diagonal is the square root of 2. If the side of a rhombus (two equilateral triangles back to back) is 1, its long axis is the square root of 3. These ratios occur all the time in the constructions of creative geometry. But they are not the most important. Above them is the primary relationship, between the diameter of a circle and its circumference. That ratio is known by the Greek letter pi. If you multiply the diameter by pi you get the circumference. But what exactly is the value of pi? There is the catch, because it turns out that pi is 'irrational'. It cannot be expressed as a ratio between two whole numbers. It is 3.141592. . . going on forever. And it is the same with other important ratios in geometry. They too are irrational. This means we cannot hope to make a perfect plan of the universe, rationally and in whole numbers. We have to use close approximations to the ratios. For pi we use 22/7 or 864/275; for the square root of 2, 99/70 and for the square root of 3, 71/41.

Next week, I will show you why these ratios were so important to the ancients, who saw them as the signature of a benevolent Creator who used the same basic rules of geometry, over and over again. Then, we shall attempt to understand how the world was really made.

**PYTHAGORAS THE MYSTIC WITH A TERRIBLE SECRET**

Pythagoras was no ordinary mathematician. He saw his work as a way of getting to the mind of the creator. He led a special cult which you could join only if you swore to become celibate and vegetarian. Members would be shown the secret of the “Prime Numbers” - which are divisible only by themselves. To reveal these to a non-initiate was a crime punishable by death! Yet all we remember him for today is his famous theorem. This applies only to right-angle triangles. It states that "the square on the hypotenuse is equal to the sum of the squares on the other two sides”. The hypotenuse is the longest side of the triangle, opposite the right angle. If you draw a square upon it, and squares upon the two smaller sides, the area of the largest square will equal the area of the two others combined.

**FRIDAY JULY 11**

Dancing Stars - Venus and Mars by Bernard Fitzwalter

**DANCING STARS!**

by Bernard Fitzwalter

Venus and Mars are at exact trine today, That’s good news for all who have love on their mind... or in their heart. The romance planet and the passion planet have a natural affinity but they are never together in the same part of the sky twice. Their movements resemble those in a rock 'n' roll jive.Imagine a couple dancing: first he holds his partner close, then she twirls away from him, until they face each other again holding hands at arms' length; he takes a step back as she comes round in front of him, and with another twirl she is back in his arms - all without their hands parting. That's exactly what Venus and Mars are doing this summer. Today they are at trine. They are standing close, in harmony.

Venus then moves on and away, and Mars stays pretty much where he is. By August 26 they will be facing each other across the dance floor of the heavens. Mars will be in reverse by then, so he will be taking a step back while Venus comes round to make another trine, this time from Scorpio (Wow! Venus in Scorpio and Mars in Pisces - that really is sexy, not so much rock 'n' roll but rose-in-the-teeth tango; we must talk more about that nearer the time) and that puts them back in each other's arms again.

So if you now yearn to sweep someone on to the dance floor... or have your world filled with candlelight and flowers, be encouraged. There’s much cosmic opportunity in the months ahead.

**TRINE SIGNALS**

A “Trine” may sound like an Australian railway engine but, in astrology, it is an angle between two planets of 120 degrees. We call it a trine because it TRIsects the sky - cutting the zodiac into three equal sections. The ancients, as John Michell explained yesterday, considered triangles powerful. Though a trine is not a right-angled triangle, it can often inspire people to stop looking at life from the wrong angle!

**LOOKING FOR MARS**

Venus is due to be close to the sun for most of the summer, so she will be hard to see. Mars, though, is now very visible, late at night. The easiest way to spot the red planet, if you are not sure what you are looking for, is to wait till next Thursday. Around midnight, the Moon will rise in the east and Mars will be next to it on the left.

**THE WORLD GOES MAD AS THE MOON GROWS FULL...**

There’s a lot happening in the sky over the next seven days, as our chart explains. Today’s entry highlights the trine that Bernard talks about in his article above. Sunday’s shows the exact moment of the Full Moon. Take a look at Thursday, when Jupiter and Venus form a happy alignment. That’s a clear sign of good fortune for someone. In coming weeks we’ll explain more about how to read these symbols. And of course, if you want to know what they mean to you, just read your daily forecast!

**SATURDAY JULY 12**

Full Moon this weekend

Hang out your garlic. Lock up your werewolf. Dust down your broomstick. There’s a Full Moon this weekend. It doesn’t ‘culminate’ till Sunday evening... so there’s plenty of time for excess, extravagance and expression. Expect hidden secrets to surface soon.**Happy Birthday!**

Nelson Mandela (Cancer born July 18, 1918) Are Cancerians patient or passionate? It’s a bit of both. Once their hearts are fired with faith, they never drop their commitment to a person or a cause. If they need to act, they’ll act. If they need to wait, they’ll wait. Nelson Mandela turns 85 on Friday. The Apartheid regime kept him in gaol for years. But he just kept on doing what he had to do.

Happy birthday also to: Today July 12; Bill Cosby (66), Cheryl Ladd (52) and Christine McVie (60), Tomorrow July 13; Harrison Ford (61), Roger McGuinn (61) and Patrick Stewart (63), Monday July 14; Gerald Ford (90) and Harry Dean Stanton (77), Tuesday July 15; Linda Ronstadt (57) and Forest Whitaker (42), Wednesday July 16; Desmond Dekker (62) and Ruben Blades (55), Thursday July 17; Camilla Parker Bowles (56), Geezer Butler (54), David Hasselhoff (51) and Donald Sutherland (69), Friday July 18; Richard Branson (53) and John Glenn (82).