Context : The projection factor p is the key quantity used in the Baade-Wesselink ( BW ) method for distance determination ; it converts radial velocities into pulsation velocities .
Several methods are used to determine p , such as geometrical and hydrodynamical models or the inverse BW approach when the distance is known .

Aims : We analyze new HARPS-N spectra of \delta Cep to measure its cycle-averaged atmospheric velocity gradient in order to better constrain the projection factor .

Methods : We first apply the inverse BW method to derive p directly from observations .
The projection factor can be divided into three subconcepts : ( 1 ) a geometrical effect ( p _ { \mathrm { 0 } } ) , ( 2 ) the velocity gradient within the atmosphere ( f _ { \mathrm { grad } } ) , and ( 3 ) the relative motion of the optical pulsating photosphere with respect to the corresponding mass elements ( f _ { \mathrm { o - g } } ) .
We then measure the f _ { \mathrm { grad } } value of \delta Cep for the first time .

Results : When the HARPS-N mean cross-correlated line-profiles are fitted with a Gaussian profile , the projection factor is p _ { \mathrm { cc - g } } = 1.239 \pm 0.034 ( stat . ) \pm 0.023 ( syst . ) .
When we consider the different amplitudes of the radial velocity curves that are associated with 17 selected spectral lines , we measure projection factors ranging from 1.273 to 1.329 .
We find a relation between f _ { grad } and the line depth measured when the Cepheid is at minimum radius .
This relation is consistent with that obtained from our best hydrodynamical model of \delta Cep and with our projection factor decomposition .
Using the observational values of p and f _ { \mathrm { grad } } found for the 17 spectral lines , we derive a semi-theoretical value of f _ { \mathrm { o - g } } .
We alternatively obtain f _ { \mathrm { o - g } } = 0.975 \pm 0.002 or 1.006 \pm 0.002 assuming models using radiative transfer in plane-parallel or spherically symmetric geometries , respectively .

Conclusions : The new HARPS-N observations of \delta Cep are consistent with our decomposition of the projection factor .
The next step will be to measure p _ { \mathrm { 0 } } directly from the next generation of visible interferometers .
With these values in hand , it will be possible to derive f _ { \mathrm { o - g } } directly from observations .