Taken from The Gospel Standard
There were little need of promises and such other encouragements as the Word of God is full of, if believers were not brought into such straits and difficulties as continually to need them. Our heart is full of unbelief, infidelity, worldliness, pride, presumption, hypocrisy, and every other hateful sin. Where these evils exist they will manifest themselves, and it is this warfare between flesh and spirit which makes true religion such a continual scene of changes. What deadness is often felt, what darkness of soul, what coldness and hardness of heart, what disinclination, yea what aversion to the things which belong to our peace! Thus we cry out, “Woe is me! My leanness, my leanness! Woe unto me! My soul cleaveth to the dust; quicken Thou me according to Thy Word. Wilt Thou not revive us again, that Thy people may trust in Thee?” Some of God’s people are not exposed to gross temptations, but as Newton says of himself, they suffer by sap and mine. The fortress of their heart, that is, is not assailed by storm, but gradually undermined by the slower process of coldness, deadness, disinclination to spiritual things, and a miserable, careless, carnal, worldly, slothful state, which benumbs all the spiritual faculties.
by J.C. Philpot
Note the phrase above — sap and mine. I asked the people at The Gospel Standard about that phrase.
Thank you for your enquiry. In simple terms, it means the insidious or slow undermining of belief. The expression is derived from soldiers making trenches (saps) or tunnels (mines) to cover their approach to a city wall, besieged place, or enemy trenches. Little by little the soldiers (Sappers) progress forward, without detection, until they destroy or undermine the object of their conquest.
Caleb Pearce, The Gospel Standard