$4-13.$ A vertical angle is measured to a signal on top of a hill. The vertical circle reads $+13{14}^{\text{'}}$ with the telescope direct and $+13{19}^{\text{'}}$ with the telescope reversed. If the vernier reads zero when the transit is truly leveled and the telescope bubble is centered, what causes the apparent discrepancy? What is the size of the actual discrepancy?

$4-14.$ In laying off a vertical angle, the surveyor sights the transit in the direction of the point to be established. He brings the telescope to a horizontal position and centers the telescope bubble. The vertical circle now reads $-0{03}^{\text{'}}.$ What reading should be set on the vertical circle to establish a $+15{20}^{\text{'}}$ vertical angle?

$4-15.$ A theodolite measures zenith angles in which the circle reads $0$ with the telescope pointed to the zenith. With the telescope direct, the vertical circle reads $129{42}^{\text{'}}{52}^{\text{'}\text{'}}$ when the line of sight is directed to a target. With the telescope reversed, the vertical circle reading is $230{17}^{\text{'}}{16}^{\text{'}\text{'}}.$ What is the zenith angle? What is the vertical angle?