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1] to construct it first draw a line of 1 unit AB 
2] perpendicular to it draw a line of 1 unit BC
3] join AC and it will be root 2 
4] then draw line perpendicular to AC as CD 
5] join the AD continue this process upto root 13.
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To construct a square root spiral follow the following steps-
1) Draw a line segment 1 cm AB and from B draw perpendicular line of 1 cm C from AB.
2) Join C and A, this is  \sqrt{2}
3) Now taking AC as base draw perpendicular and continue the same process as above.
For help i am drawing this....
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What is this you have draw clearly