E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
To burn the candle at both ends. To be like a man on double business bound, who both neglects. Of course, no candle could burn at both ends, unless held horizontally, as the lower end would be extinguished by the melted wax or tallow.
To make two or both ends meet. To make ones income cover expenses; to keep out of debt. The allusion is to a belt somewhat too tight. The French say joindre les deux bouts.