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The paper contains a proof, that if every number of the form 8n + 4 is composed of 4 odd squares, then every number whatever must be composed of 4 square numbers or less; also a proof of the converse of this, viz. that if every number is composed of 4 square numbers or less, then every number of the form 8n + 4 must be composed of 4 odd squares. It is then proposed to show that every number of the form 8n + 4 is composed of 4 odd squares, by taking a number of the form 8n + 4, viz. an odd square +3, and showing that 8n + 4 in that case is divisible into 4 odd squares (other than the odd square and 1, 1, 1); thus 16n 2 ± 8n + 1 is a form that includes every odd square, and 16n 2 ± 8n + 4 is divisible into 4n 2 ± 4n + 1, 4n 2 ± 4n + 1, 4n 2 ± 4n + 1, 4n 2 ∓ 4n + 1.

A proof (by means of a series) that every number is composed of 4 square numbers, or less, without reference to the properties of prime numbers